Intersecting hypersurfaces and Lovelock Gravity - arXiv

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arXiv:gr-qc/0502089v1 22 Feb 2005

Ph.D. Thesis.

Intersecting hypersurfaces and Lovelock Gravity Steven Willison1 Department of Physics, Kings College, Strand, London WC2R 2LS, U.K.

July 2004

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E-mail: [email protected]

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Abstract A theory of gravity in higher dimensions is considered. The usual Einstein-Hilbert action is supplemented with Lovelock terms, of higher order in the curvature tensor. These terms are important for the low energy action of string theories. The intersection of hypersurfaces is studied in the Lovelock theory. The study is restricted to hypersurfaces of co-dimension 1, (d − 1)-dimensional submanifolds in a d-dimensional space-time. It is found that exact thin shells of matter are admissible, with a mild form of curvature singularity: the first derivative of the metric is discontinuous across the surface. Also, with only this mild kind of curvature singularity, there is a possibility of matter localised on the intersections. This gives a classical analogue of the intersecting brane-worlds in high energy String phenomenology. Such a possibility does not arise in the pure Einstein-Hilbert case.

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Acknowledgements I would like to thank N. E. Mavromatos, my supervisor, for valuable advice and encouragement; E. Gravanis for great fun working together; my parents for constant support and understanding. I thank God for all the opportunities I have had. Thanks to CERN Theory department for their hospitality and to the European Union for funding the trip through Network ref HPRN-CT-2000-00152. The Ph.D. was funded by EPSRC, to whom I am very grateful.

The material in Chapters 4, 5 and 6 represent the original work of Elias Gravanis and myself as appearing in J. Math. Phys. 45, 4223 (2004) [37] and gr-qc/0401062 [38] with emphasis on my own contributions. Some of the material in Chapter 3 is my own original previously unpublished work. The rest is a survey of the literature and is fully cited.

CONTENTS

1. Introduction . . . . . . . . . . 1.1 Extended objects . . . . 1.2 Extra dimensions . . . . 1.3 Thin walls and gravity . 1.4 Higher curvature gravity 1.5 Summary of contents . .

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4. Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Euler densities . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 From boundary to intersection action terms . . 4.1.2 Manifolds with discontinuous connection 1-form 4.2 Dimensionally continued action . . . . . . . . . . . . .

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48 49 53 53 54 55 57 60 60

2. Einstein and Lovelock Gravity . . . 2.1 General Relativity . . . . . . 2.2 Lovelock gravity . . . . . . . . 2.3 Orthonormal frames . . . . . 2.4 Some notation . . . . . . . . . 2.5 Dimensionally continued Euler 3. The 3.1 3.2 3.3 3.4

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junction conditions . . . . . . . . . . Hypersurfaces . . . . . . . . . . . . . Distributional curvature . . . . . . . Gravity actions and boundary terms Gluing manifolds together . . . . . .

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5. Homotopy parameters and simplices . . . . . . . . . . . . . . . . 5.1 A geometrical approach . . . . . . . . . . . . . . . . . . . . . 5.2 The implications . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Dual simplices . . . . . . . . . . . . . . . . . . . . . . 5.2.2 F-space and Homotopy Operator . . . . . . . . . . . 5.2.3 Non-simplicial intersections made simple . . . . . . . 5.3 Dimensionally continued Euler density . . . . . . . . . . . . 5.4 The interpolating curvature . . . . . . . . . . . . . . . . . . 5.5 Intersecting hypersurfaces in more general theories via closed

Contents

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7. Concluding remarks and outlook . . . . . . . 7.1 Intersecting or colliding braneworlds . . 7.2 Acceptable singularities . . . . . . . . . . 7.2.1 A question of derivatives . . . . . 7.2.2 Conical singularities and the like 7.2.3 Singular gravity sources . . . . . 7.2.4 Gravity on simplicial manifolds . 7.3 Geometrical conundrums . . . . . . . . . 7.3.1 Non-simplicial intersections . . . 7.3.2 Mappings . . . . . . . . . . . . . 7.3.3 The dual lattice . . . . . . . . . .

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6. An 6.1 6.2 6.3 6.4

explicit example . . . . . . . . . . . . . The N hypersurface intersection . . . . Energy conservation at the intersection Colliding Branes and deficit angles . . A three-way intersection in AdS . . . . 6.4.1 The bulk vacuum solution . . . 6.4.2 Three-way intersection . . . . .

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Appendix A. Some mathematical preliminaries . . . . . . . . . . . . . . A.1 Extrinsic curvature . . . . . . . . . . . . . . . . . . . A.2 Exterior differential calculus . . . . . . . . . . . . . . A.2.1 Orthonormal frames . . . . . . . . . . . . . . A.2.2 Integration of forms on oriented manifolds . . A.2.3 Integration over simplices and chains . . . . . A.3 The Euler number . . . . . . . . . . . . . . . . . . . A.3.1 A useful property of the invariant polynomial A.4 Honeycombs . . . . . . . . . . . . . . . . . . . . . . .

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B. Useful formulae . . . . . . . . . . . . . . . . . . . . B.1 The variational principle for a gravity theory . B.2 The variational principle for Lovelock gravity . B.3 Second fundamental form and tensors . . . . . B.4 Decomposing the bulk Lagrangian . . . . . . . B.5 Proof of dη = 0 . . . . . . . . . . . . . . . . .

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C. Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

1. INTRODUCTION The General Theory of Relativity was a great conceptual advance and a great piece of twentieth century physics. The rigorous pursuit of the principle of the invariance of the laws of physics led to the idea that space-time was a dynamical manifold. This brought into physics the abstract study of non-Euclidean geometry by mathematicians such as Riemann. It also elevated space and time to the status of a dynamical structure. The application of geometry to physics also plays an important role in gauge theories. The possibility to represent gauge theories in terms of connections on fiber bundles has been of great use. The latter kind of geometrical theory has proved to be very much compatible with the principles of quantum mechanics. The discovery that non-abelian gauge theories are re-normalisable allowed the description of the electro-magnetic, weak and strong nuclear forces in terms of a re-normalisable quantum theory, the Standard Model of particle physics. Gravity has proved much more difficult to quantise. A major goal of theoretical physics is to find a consistent quantum theory of gravity. The canonical approach of quantisation to General Relativity leads to a non-renormalisable theory. Various approaches to the problem of quantum gravity have been attempted. String theory conceives of extended particles in a flat background. The modes of the strings give rise to gravitons as well as various other fields. Loop quantum gravity is a background independent theory which takes very seriously the principles of GR. The Ashtekkar variables have allowed for the treatment of GR as a gauge theory. There are also approaches, motivated by work in condensed matter physics, where the properties like general coordinate invariance are emergent properties. In this case the underlying theory is not important - only the quasi-particle spectrum. Other approaches are simplicial quantum gravity, Chern-Simons theories, non-commutative geometry. The list of proposed quantum theories of gravity seems to be endless. Also, it has been proposed that quantum gravity may actually be a deterministic theory [81].

1.1 Extended objects The idea of fundamental particles as point particles is conceptually very tidy. To date, experiment has not revealed any substructure to certain particles such as the electron. The idea of matter as continuous and infinitely sub-divisible has also enjoyed the favour of scientists at various times in history. Einstein thought that a fundamental particle would be a stable, smooth, matter configuration. There are no-go theorems prohibiting such solitons in GR and Einstein-Maxwell theory. Some smooth solitonic

1. Introduction

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Too many dimensions

!?!? Horrible maths

Implausible scenarios

Non−existant particles

Not enough dimensions

Fig. 1.1: Many roads to quantum gravity

solutions have been found in more general Einstein-Yang-Mills theories [6]. Another possibility is particles not as points but membranes. In the last thirty years there have been a number of advances in theories of extended particles featuring extra dimensions. Type I strings, Type IIa and IIb strings, Heterotic E8 × E8 and Heterotic SO(32) strings all live in ten dimensions. These theories are related to each other and also to the eleven dimensional super-gravity by dualities and are believed to be part of an underlying theory called M-Theory [73]. A feature of strings and eleven dimensional super-gravity is the existence of membrane solutions. They are solitons of the super-gravity theories. In open string theories they are p-dimensional surfaces on which open strings can have their end-points.

1.2 Extra dimensions Another common feature, as already mentioned, is extra dimensions. The interest in extra dimensions long predates string theory. As early as the 1920’s people looked into extra dimensions to generate a gauge field. The Kaluza-Klein compactification of a fifth dimension on a circle gives a photon type field. It was noticed later that there is also a scalar field which can not be ignored. We are familiar with three dimensions of space and one of time, but the string/M theory tells us we should have another six or seven dimensions lurking about somewhere. It was thought that if such a theory is to be taken seriously, these extra dimensions must be compactified very tightly else they would already have been ruled out by experiment. Questions as to why the theory should compactify down to four dimensions arise. This is part of a more general problem of fine tuning. Many physical parameters seem to be finely tuned to be amenable to life. e.g. the stability of atoms in four

1. Introduction

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dimensions. Anthropic arguments can be invoked, whereby the seeming fine tuning of various physical quantities for life is not viewed as fundamental but just a part of the complicated geography of the universe. We just live in a part that happens to be hospitable for life. But an explanation in terms of a unique solution to a fundamental theory is desirable to many people. Consequently, a dynamical unique compactification to four dimensions has been sought after. Lately, the idea that we live confined to a membrane in higher dimensions has been suggested as in the much hyped Randall-Sundrum case [74, 75] and a similar earlier idea by Rubakov and Shaposhnikov [77]. It is interesting that certain branes in string theory necessarily have gauge fields localised on them. Gravity, as a closed string mode would be free to propagate in the higher dimensions, but if they are compactified tightly enough, the deviations from the inverse-square law are small enough to evade current experimental falsification. As such, they constitute some kind of prediction for future gravity experiments. Another possibility is that gravity can be effectively localised even with large extra dimensions as demonstrated by Randall and Sundrum. Another interesting phenomenological feature of strings is the Standard-Modellike effective field theories of intersecting D p-Branes. The open strings stretching between the intersecting stacks of branes can be localised around the intersection due to the tension of stretching them. In particular, there are Chiral matter fields living on the intersection [8]. Thus, around a 3+1 dimensional intersection there is potentially a realistic standard model low energy physics.

1.3 Thin walls and gravity I will consider the role of membranes in higher dimensions from a classical perspective. The initial motivation for such a study was phenomenological- the modeling of Braneworlds of the Randall-Sundrum type, in general gravitational backgrounds, including the branes own gravity. Brane cosmology is very popular at the moment. Presumably, this is because it offers possibilities of solving some perceived problems in cosmology and phenomenology. ∗ Brane worlds may resolve the Gauge Hierarchy problem. ∗ Problems of fine tuning such as the cosmological constant problem and the cosmic coincidence problem could be explained away. ∗ The initial singularity could be avoided and replaced by a “big bounce” or some other kind of eternal universe. The problem of initial conditions is avoided. ∗ Contact is made between String/M-theory and four-dimensional physics. The need to reduce the number of free parameters in any theory of physics is an understandable one. It would be nice to have a solution to the gauge hierarchy

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problem and have one energy scale rather than two. This is in accordance with a basic intuition on which science is based: that diverse phenomena can be explained in terms of unifying principles. The desire to avoid special initial conditions is a different matter though. This seems to represent an atheistic approach to a question which goes beyond the normal scope of Physics. This is an incorrect approach in my opinion. In general, if a manifold is filled with membranes, they will intersect in all sorts of ways. I have already mentioned intersecting branes. People have also studied colliding brane-worlds, the “Ekpyrotic” scenario [46], as a cosmological model. As well as String inspired cosmology, there are several other motivations for studying thin walls of matter. For example, they have been used to make simple models of quantum black holes. The principle of holography is also important in quantum gravity. The realisation that the degrees of freedom of a black hole live on the surface area has profound implications. The study of hypersurfaces may be relevant to this. Another situation where hypersurfaces and their intersections is relevant is in the various inflationary models. There has been much study on the behaviour of bubbles of false vacuum, in the context of phase transitions in the early Universe. There has been speculation that a bubble of false vacuum (de Sitter space) in some primordial space-time foam could have inflated to give birth to a ‘universe’. Possibly this could have happened many times, with many universes being created out of the quantum foam. The simplest approximation is to model the interface between the false vacuum and surrounding space-time as a singular wall of matter. Such a wall is known as a shell in this context. The work has been concentrated on isolated spherical bubbles, but a more general foam of bubbles would be interesting to study. There are situations in astrophysical GR where a thin wall approximation is useful, such as the ejection of matter from supernovae or the motion of globular clusters [5]. The above are concerned with speculations and approximations. But I will attempt to show that the study is profitable in its own right and will give some insights into classical gravity as a geometrical theory. In this study, I will restrict myself to surfaces of co-dimension one, which I shall call hypersurfaces or walls, and their intersections. For example, a wall in 4+1 dimensions of space and time will be a 3 + 1 or 4 dimensional surface. Clearly, the study of membranes of other co-dimensionalities is also relevant and I will comment on this in the concluding section.

1.4 Higher curvature gravity The modification of the Einstein tensor has been suggested in many contexts: counterterms in general relativity to regulate singularities; scalar-tensor theories in inflationary contexts; terms appearing in super-gravity; low energy actions from strings. GR is not a renormalisable theory. The Lagrangian has a negative dimension −2 coupling ∝ MPlanck . Each graviton self interaction term i contributes −Ni ∆ to the superficial degree of divergence, D, where Ni is the number of vertices and

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∆ = [coupling] = −2. For FeynmanR diagrams with internal closed loops, one has ∞ integrals over momenta of the form 0 dkk D−1 which diverge when D > 0. General relativity is renormalisable at one loop but, since −Ni ∆ is positive, at each order in the loop expansion, more divergences arise. This requires absorbing divergences by re-normalising an infinite number of parameters. Consequently none of its interactions are re-normalisable. Couplings to matter (and gravity couples to everything) make the divergences even more marked. Also, in higher than four dimensions ∆ = −(d − 2) and the same problems arise A non-renormalisable theory is not considered to be a reasonable candidate for a final theory. A naive approach to quantum gravity means one has to add every term, including all the non-gravitational terms, allowed by the symmetries, as counterterms. This makes the theory fairly useless at high energies from the point of view of predictive power- there are an infinite number of phenomenological parameters. This is why non-renormalisable theories tend to be regarded as effective field theories. Integrate out a massive particle φ , of mass M, and consider a low energy theory with partition function Z exp (i ∫ Leff ) = [dφ] exp (i ∫ L) at energies, E, much lower than this mass scale. The result is that Leff is composed of non-renormalisable terms suppressed by E/M to some power. For example, starting with QED, integrate out the electron/positron and you get non-renormalisable theory for photon-photon scattering [86]. Similarly, GR can be viewed as the low energy effective theory of some unknown fundamental quantum theory which may be very different at the Planck energy and length scale. Certain theories of gravity with curvature tensor squared terms have been suggested to render gravity renormalisable in four dimensions [79]. Consider a general action quadratic in curvature. Z √ (1.1) (α1 −gR + α2 L2 )dd x, M √ L2 = g(Rµνκλ Rµνκλ + aRµν Rµν + bR2 ). The quadratic term has a dimensionless coupling constant, [α2 ] = 4 − d. Consequently, its graviton self interaction terms are expected to be re-normalisable in 4 dimensions. This has motivated the study of such quadratic theories. However, this approach to the re-normalisation problem proved to be a dead-end due to a loss of unitarity of the S-Matrix. Adopting the covariant perturbative approach to quantisation, one expands about a flat metric1 g µν = η µν + hµν . Then the field hµν is treated like a spin 2 field in Minkowski space. The Einstein-Hilbert term contains only two derivatives, the quadratic term contains four. The higher derivatives dominate at high frequency and potentially render the theory re-normalisable. The higher derivatives are also responsible for the appearance of ghosts. To see the problem, 1

The expansion has only a finite number of terms, so h need not be small. The only assumption is that space-time has topology Rd and is asymptotically flat.

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one need look at the propagator. Let us do the calculation. R = Γµνν,µ − Γµνµ,ν + Γµλµ Γλνν − Γµλν Γλνµ , 1 Γαβγ = (η + h)αλ (hβλ,γ + hλγ,β − hβγ,λ ). 2 The comma denotes partial differentiation. The diffeomorphism invariance allows one . to impose the harmonic gauge fixing condition hµν, ν = 12 h, µ . Let = mean equality in harmonic gauge. √

3 . 1 −gR = − hµν hµν − hh + ∂(· · · ) + O(h3 ). 4 8

There will be additional quadratic terms in hµν coming from the curvaturesquared terms in (1.1) [92]: . L2 = (a + 4)hµν 2 hµν + (b − 1)h2 h + ∂(· · · ) + O(h3 ). The higher order terms in the field hµν do not affect the propagator. They will contribute to the self interaction of gravity in perturbation theory. The propagator is: n o −1 1 3 −α1 ηµα ηνβ + ηµν ηαβ + α2 (a + 4)ηµα ηνβ + (b − 1)ηµν ηαβ 2 . 4 8 It is important to note that the fourth derivative contribution to the propagator vanishes if and only if a = −4, b = 1. In four dimensions, there is the Gauss-bonnet identity: The term R2 − 4Rµν Rµν + Rµναβ Rµναβ

(1.2)

is actually a total derivative and does not contribute to the local degrees of freedom of the theory at all. In higher dimensions, this term is not a total derivative. It contributes to the graviton self-interactions but not the propagator. In momentum space, the components of the propagator are of the form k2

−1 + λk 4

(1.3)

with λ ∼ α2 /α1 . If we expand this as: −1 λ + , 2 k 1 + λk 2

(1.4)

then whatever the sign of λ, the k 2 in the second term comes with the wrong sign. This wrong sign indicates that there are propagating ghosts in the quantum theory. The appearance of ghosts is a quite generic feature of higher derivative theories. To get more insight into this, consider for simplicity a scalar field instead of a spin 2 field. A theory with higher derivatives (φ, φ, 2 φ) can always be re-cast as second order by field redefinitions, say ψi := ψi (φ, φ), so that the fields are (ψi , ψi ).

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The higher derivatives introduce extra particles. For the higher derivative gravity, it turns out that there is always a massive graviton with the wrong sign in the propagator, which spoils the physical interpretation of the theory. Only the GaussBonnet combination introduces no extra particles. There is just the massless graviton of the normal theory. Consider the (scalar) field theory described by the higher derivative Lagrangian: L = α1 φφ + α2 φ2 φ + (interactions). The “kinetic” term is: α2 φ( + α1 /α2 )φ. To bring this into a second derivative form, make the field redefinitions: ψ1 =

α2 φ, α1

ψ2 =

α2 ( + α1 /α2 )φ. α1

Then, up to total derivatives: α2 φ( + α1 /α2 )φ = α1 (ψ2 − ψ1 )( + α1 /α2 )ψ1 = α1 (ψ2 − ψ1 )ψ2 α2 = 1 ψ1 ψ2 . α2 Taking a linear combination of these, we get, up to total derivatives, for the kinetic terms: α1 ψ2 ψ2 − α1 ψ1 ( + α1 /α2 )ψ1 . We have a massless particle plus another, massive particle. This massive particle inevitably comes with a minus sign in the kinetic term relative to the massless particle. The interaction terms will be products of the linearly independent combinations φ = ψ2 − ψ1 ,

φ =

α1 ψ1 . α2

The two particle species will interact with each other. Classically, there are two fields, one with positive energy and one with negative. Although the choice of sign is arbitrary, the relative minus sign is significant, the interactions causing both fields to grow exponentially. The same argument applies for the spin 2 field. From the point of view of our perturbative expansion of the metric about flat space-time, it means that Minkowski space is unstable. If ψ1 , ψ2 are quantum fields, the negative energy interpretation can be avoided by choosing ψ2 to be a ghost field, a boson which has Poisson bracket replaced by anti-commutator instead of commutator. Then, the energy is positive but a state representing an odd number of particles has negative norm. The negative probabilities make the physical interpretation very problematic. The violation of the spin-statistics law means that either the S-Matrix is non-unitary or the principle of local causality

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is violated. These are two very important physical principles underlying quantum field theories. We are left with two cases: 1) Gauss-Bonnet combination: non-renormalisable but makes sense as a low energy effective theory, Minkowski space is stable. 2) Any other curvature-squared combination: In principle re-normalisable in four dimensions2 , but inevitable presence of ghosts makes the theory unphysical. In four dimensions, the Gauss-Bonnet combination is locally a total derivative. It does not affect the local degrees of freedom. Thinking now of the path integral approach to quantisation, it merely contributes a topology-dependent weight to the functional integral. In higher dimensions, although the “kinetic term” is a total derivative, the GaussBonnet term does contribute to the self-interactions of the graviton. In higher dimensions then, the Gauss-Bonnet is the only curvature squared term, containing the metric and its derivatives, which does not suffer from ghosts, and furthermore, the particle content is the same as GR: a massless spin two particle. It is worthwhile to look at modifications to GR. New gravitational physics may have something to say on: the cosmological constant problem; the dark matter problem; black holes and the problems associated with them such as unitarity, entropy [65]. A particularly interesting higher curvature theory is the Lovelock gravity. This is a generalisation of General relativity in a special sense. The Einstein-Hilbert action is the dimensional continuation of the two-dimensional Euler Characteristic. The Gauss-Bonnet term (1.2) is similarly related to the 4-dimensional Euler Characteristic. The general Lovelock action is a sum of terms which are similarly the dimensional continuation of Euler Characteristics of each even dimension. Aside from a cosmological constant, the other Lovelock terms are only non-zero in higher than four dimensions. If we are looking for terms which are some low energy expression of a fundamental quantum theory of gravity in higher dimensions, such as String Theory, which is unitary and which has flat space as a vacuum, the Gauss-Bonnet term is a natural one to consider. Thinking of purely gravitational terms, other curvature-squared combinations would not be expected to arise [92, 2]. However, there would be other fields coupling to gravity in various ways. Also, generically, curvature cubed terms and higher do not contribute to the propagator. The cubic Lovelock term is not special in this sense. There are many nice properties related to the geometrical origins and quasilinearity property which will be discussed in chapter 2, for example, the Cauchy problem is very similar to GR. The Lovelock theories have been studied extensively. Higher dimensional black holes have been studied [11, 3, 13, 21]. This has shed some interesting light on 2

This is for pure gravity. Couplings to matter fields would not be expected to be re-normalisable.

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questions of black hole entropy. Some cosmological metrics have been studied [55, 24]. In particular, the Lovelock contributions, motivated by string theory, have played a big role in studying braneworld cosmology3 . Such higher curvature terms, as well as other stringy fields like dilaton and moduli, have led to all sorts of possibilities. It has been suggested that the Gauss-bonnet gravity can account for cosmological expansion without the problem of missing dark matter [23]. Localisation of gravity in large extra dimensions has also been realised in this context. The discovery by Chamseddine [14] that certain Lovelock theories are equivalent to Chern-Simons gauge theories of the de Sitter or anti-de Sitter gauge group has also generated interest [90]. Apart from anything else, as a mathematical theory with close connections to interesting geometry, the Lovelock theory deserves some attention. It should be stressed that the motivations for study are all theoretical. There is no experimental evidence as yet for such modifications to gravity. The special relevance of Lovelock terms to intersecting braneworlds was realised by Kim, Kyae and Lee [47, 48]. Very recently after the work presented here was first announced, intersecting brane-worlds in pure Lovelock gravity have been studied [52, 67] by others and an example has been found of a 4-dimensional intersection in 10 bulk dimensions. I hope that this work, being a general treatment of intersections, will be useful in such studies.

1.5 Summary of contents In the second chapter I will discuss the Lovelock Theories of Gravity in higher dimensions. In the third chapter the singular hypersurfaces method of Israel is introduced. The notion of distributional curvature is discussed along with the problems with extending these concepts to the higher curvature theories. The study of boundary terms in gravitational actions is also introduced. The possibility of using boundary terms in the Lagrangian to describe membranes is discussed and the method is compared to integrating the field equations over distributional sources. In the fourth chapter I present the results of [hep-th/0306220] co-authored with E. Gravanis. First, I deal with the Euler density of manifolds containing a certain kind of discontinuity in the geometry. We found a chain of terms, starting with the Euler density in 2n dimensions, obeying a simple relation: A sum of terms gives the total derivative of another term. By a simple inductive argument, it is shown that these terms ‘live’ on the intersection of hypersurfaces of discontinuity in the geometry. I then generalise to the dimensionally continued Euler density. The reason for studying the discontinuities is made clear. It corresponds to the Israel type junction and distributional sources. We show that it is possible to construct an action which 3

some of the many works are [59, 25, 15, 34, 60, 22, 39]

1. Introduction

15

gives all the junction conditions for the intersections, as well as the gravitational field equations. In the fifth chapter I present the results of [gr-qc/0401066], again co-authored with E. Gravanis. The problem is re-formulated in terms of the simplices dual to the intersections. Again turning to the dimensionally continued Euler density, the action constructed in chapter four is further justified by Lemma 5.6. Chapter six has examples illustrating the possible physical applications. In the final chapter, the possibilities for future research and possible interpretations of the geometrical features are discussed. Mathematical details, definitions of simplices and a glossary are in the appendices.

2. EINSTEIN AND LOVELOCK GRAVITY 2.1 General Relativity Gravity, via the equivalence principle, leads to relative acceleration between local Lorentz frames at different places. In other words, space-time is curved. There is no global inertial frame. We have to do away with rigid Minkowski space in favour of a dynamical manifold. In particular, space-time is no longer a vector space. There is no meaningful way to define or add displacement vectors. Instead vectors live on the tangent bundle, corresponding to infinitesimal directional derivatives. Matter and (non-gravitational) energy, in the form of the Energy-momentum tensor is the source of the curvature via the famous Einstein Equation. This is a form of Mach’s principle, the definition of inertial frames being dependent on the matter-energy content of the universe. The Einstein Equations are Tµν = Gµν

(2.1)

or, including the cosmological constant, Tµν = Gµν + Λgµν ,

(2.2)

where Gµν is the Einstein tensor constructed solely from the metric tensor and it’s derivatives: 1 (2.3) Gµν = Rµν − Rgµν . 2 gµν is the metric which gives us a measure of distance. Rµν is the Ricci Tensor. It is α 1 derived from the Riemann curvature tensor Rµβν by contraction of first and third indices. The Riemann tensor is constructed from the metric compatible, torsion-free connection coefficients Γαβγ . Γαβγ are known as the Levi-Civita connection coefficients or Christoffel Symbols. The metric compatible connection defines the parallel transport of vectors such that their metric products (norm and angle) are preserved. The Riemann tensor measures the non-commutativity of the associated covariant derivative. R = Rµµ is the Ricci scalar. Einstein’s equations (with cosmological constant) obey three very important principles [91]: 1) they are independent of reference frame determined by choice of co-ordinates, 2) there is well defined Cauchy problem for the evolution of the metric tensor,2 1 2

α α λ I use the convention Rµβν ≡ ∂β Γα µν + Γλβ Γµν − (β ↔ ν) for globally hyperbolic space-time (see Wald Ch 8,10 [85])

2. Einstein and Lovelock Gravity

17

3) they reduce to Newtonian Gravity in the weak field, non-relativistic case. Condition 1 is satisfied because Einstein’s equations constitute a tensor relation. Condition 2 means that the Cauchy conditions are necessary and sufficient to integrate the vacuum equations. Specifying the field and its first derivative on an initial Cauchy surface will fully determine the time evolution of the field. This means that there exists a Hamiltonian formulation. If higher derivatives need to be specified, one can always redefine them as new variables and the theory is again second order. But we then have new observables and the theory is qualitatively different from the Newtonian physics. The form of the Einstein tensor ensures that the metric has a well defined Cauchy problem. Whether the whole Einstein equations have such a well-defined initial value problem will depend on the matter content. Condition 3 is in accord with measurement from everyday physics to celestial mechanics that all non-relativistic, classical, weak gravity systems obey Newtonian physics. As an example consider a spherically symmetric Schwarzschild solution around a spherical mass distribution. The weak field condition means r/r0 >> 1 where r is the radial co-ordinate and r0 is the Schwarzschild radius. In this limit, the Newtonian physics is indeed recovered [85]. If we demand in addition that Minkowski space be a vacuum solution then the Cosmological term is eliminated (or very small). The Einstein tensor (2.3) is the only tensor that is: A) Symmetric, B) Covariantly conserved: ∇µ Gµν = 0, C) depending only on the metric, its first and second derivatives, D) linear in second derivatives of the metric. As such it is a suitable ingredient for the field equations and obeys the three physical conditions above. The importance of conditions (A) and (B) is apparent since an energy-momentum tensor, coming from the variation of a matter lagrangian with respect to the metric, is both symmetric and divergence free. These conditions are automatically satisfied by the existence of the Lagrangian for gravity. In other words, if the whole mattergravity system comes from a lagrangian, then conditions (A) and (B) are satisfied for both sides of the Einstein equation. The action, which yields the Einstein field equations, is known as the Einstein-Hilbert action: Z √ S= d4 x g R, (2.4) M

g ≡ det gµν . Condition (B), the contracted Bianchi Identity, gives the local conservation of energy and momentum.


18

Conditions (C) and (D) are important for the physical conditions (2) and (3). This second order condition is not absolutely essential if we are describing an effective theory for the low energy of some more fundamental theory. However, the second order field equations are important for the classical stability of solutions and are the most natural choice by analogy with classical mechanics.

2.2 Lovelock gravity In higher dimensions, there are other tensors admissible if condition D is relaxed to quasi-linearity [26]. The definition of quasi-linearity is basically that there are no squared or higher order terms in second derivatives of the metric with respect to a given direction. So for our higher dimensional theory we have the following reasonable conditions: i) The classical equations of motion are equivalent to a principle of extremal action; ii) The Euler variation with respect to the metric produces a tensor depending only on the metric, its first and second derivatives; iii) The tensor is quasi-linear in second derivative terms. The tensor quadratic in the Riemann tensor was found by Lanczos. All such tensors were found by Lovelock [54] along with the corresponding action. [d/2]

Hνµ S=

=−

[d/2] Z

X n=0

X 1 ν2n β δ µµ1 ...µ2n Rµν11νµ22 · · · Rµν2n−1 , 2n−1 µ2n n+1 n νν1 ...ν2n 2 n=0

M

1 µ1 ...µ2n ν1 ν2 ν2n−1 ν2n √ β δ R · · · R g dd x. n ν ...ν µ µ µ 1 2n 1 2 2n−1 µ2n n 2

(2.5)

(2.6)

The delta is the generalised totally anti-symmetrised Kronecker delta. It is the M determinant of a matrix with elements δN , µ

µ1 ...µp M δνµ11...ν = det(δN ) = p!δ[ν δ p p 1 νp ]

with the index M running from µ1 , ..., µp and likewise N = ν1 , ..., νp . The n = 0 term in (2.6) is the Cosmological constant. The n = 1 term is the Einstein-Hilbert term. The n = 2 term is proportional to: R2 − 4Rµν Rµν + Rµναβ Rµναβ and is variously known a Gauss-Bonnet term, Lanczos term or Lovelock term. The quasi-linearity can be seen from the totally anti-symmetric form of the Kronecker delta in (2.5). The terms containing products of second derivatives of the


19

metric are of the form: µνκλ... δαβγδ... ∂µ Γαβν ∂κ Γγδλ · · ·

The anti-symmetry means that the derivatives are in orthogonal directions. One of the consequences of quasi-linearity w.r.t. the second time derivatives is that the Lovelock theory is ghost free in perturbation theory about a flat background [92, 93]. On the down side, there are some problems with the evolution of the classical solutions due to the multiple solutions of polynomial equations [26]. In d = 2n, the integrand √ µ1 ...µ2n ν1 ν2 ν2n gδν1 ...ν2n Rµ1 µ2 · · · Rµν2n−1 (2.7) 2n−1 µ2n goes by the name of the Lipschitz-Killing curvature. This tensor appears in the Gauss-Bonnet formula for 2n dimensions (Appendix A.3). This similarity with the Gauss-Bonnet formula explains many of the interesting properties of the theories [71, 93]. It will be especially important for our study of intersecting hypersurfaces.

2.3 Orthonormal frames Rather than working with tensors, we will use the method of exterior differential forms. The method replaces metric and Christoffel symbol degrees of freedom with orthonormal frame (or vielbein) and spin connection (or connection 1-form). This would be essential when dealing with spinors on curved space-time. We will not be interested in spinors but the formalism will be useful. It leads to quite an abstract approach. We shall see returns on this investment in mathematics in the end when the formulae will become very simple. Another reason for using frames is that we can choose to restrict ourselves to frames adapted to an embedded submanifold. The frame field over a d-dimensional manifold is the map from an orthonormal basis of sections of the tangent bundle to a basis of Rd . In English, the vielbein is a non-co-ordinate basis which provides an orthonormal basis for the tangent space at each point on the manifold. gµν dxµ ⊗ dxν = ηab E a ⊗ E b Strictly speaking E a , being one-forms, are the co-frames, but I will not distinguish. In the vicinity of an embedded submanifold of co-dimension d, we can adapt the frame field to the submanifold. Split the Minkowski space into Rd−p ⊗ Rp . Rp gets mapped to the normal bundle by the frame field. So we can choose E i to be tangential directions and E λ to be normal say, where i = 1, ..., d − p and λ = d − p + 1, ..., d. We also introduce the metric compatible Lorentz connection and associated connection one-form ωba (or spin connection). We shall make use of exterior differential calculus. Exterior differential forms and exterior calculus are reviewed in the Appendix. The action will now be built out of the vielbein and connection and their exterior derivatives. The curvature is, according to the first of Cartan’s structure equations: Ωab = dωba + ωca ∧ ω cb.

(2.8)


20

The second of Cartan’s structure equations defines the Torsion: T a = dE a + ωba ∧ E b = DE a .

(2.9)

In GR the torsion tensor is constrained to vanish. When this constraint is not imposed, we have the Einstein-Cartan theories. When the torsion is non-vanishing, parallel transport and distance are independent of each other. The exterior derivative of the structure equations gives the Bianchi identities. DΩab = dΩab + ωca ∧ Ωcb − Ωac ∧ ωbc = 0, a

a

DDE = dT +

ωba

b

∧T =

Ωab

b

∧E .

(2.10) (2.11)

The Einstein-Hilbert action can be written compactly as L = α1 Ωab ∧ E c ∧ E d ǫabcd ǫabcd is the totally anti-symmetric Levi-Civita tensor density such that ǫ0123 = +1. When finding the equations of motion, we make use of: δΩab = D(δω)ab The explicit Euler variation with respect to the connection is δω L = α1 D(δω)ab ∧ E c ∧ E d ǫabcd = d α1 δω ab ∧ E c ∧ E d ǫabcd − 2α1 δω ab ∧ T c ∧ E d ǫabcd

The first term is a total derivative and does not contribute to the equations of motion. The second term will vanish if the torsion is zero. Thus, unless there are spinors in the matter side of the action coupling to the spin connection, or the det(vielbein) vanishes, the vanishing of the torsion is actually an equation of motion. The Einstein field equation comes just from the explicit variation with respect to the vielbein. δE L = α1 δE c Ec(1) ,

Ec(1) = 2Ωab ∧ E d ǫabcd In the familiar tensor language, the Einstein Tensor comes only from the variation of the metric, where g µν and Γαβγ are varied independently. This is the Palatini formulation. The Lovelock action can also be written very compactly in the differential form language. The action is: X (2.12) L(ω, E) = βn Ωa1 ...a2n ∧ ea1 ...a2n , n

where Ωa1 ...a2n ≡ Ωa1 a2 ∧ · · ·Ωa2n−1 a2n is the n-fold wedge product of curvature twoforms and ea1 ...ar ≡

1 ǫa ...a E ar+1 ∧ · · ·E ad . (d − r)! 1 d

(2.13)


21

The Euler variation with respect to the spin connection will again vanish (up to a total derivative) if the torsion vanishes: X δω L = βn D(δω)a1 a2 Ωa3 ...a2n ∧ ea1 ...a2n n

=

X n

βn d (δω a1 a2 Ωa3 ...a2n ∧ ea1 ...a2n ) − δω a1 a2 Ωa3 ...a2n ∧ T a2n+1 ea1 ...a2n+1

but now there are also other solutions for det(e) 6= 0. If we require zero torsion it must be imposed as a constraint. Assuming this constraint, again the gravitational field equation comes from the explicit variation with respect to the vielbein: X δE L = βn δE b ∧ Ωa1 ...a2n ∧ ea1 ...a2n b . (2.14) n

We can once again reformulate our guiding principles [93, 76, 58] a) The Lagrangian is invariant under Local Lorentz transformations, constructed from vielbein, spin-connection and their exterior derivatives, without reference to Hodge duality. b) The torsion is zero. One can also generalise to a Lovelock-Cartan theory with torsion degrees of freedom [58].

2.4 Some notation Sometimes it will be more convenient to write X L(ω, E) = αn Ωa1 ...a2n ∧ E a2n+1 ...ad ǫa1 ...a2n n

where E a2n+1 ...ad ≡ E a2n ∧···E ad is the (d−2n)-fold wedge product of vielbein frames. It should be remembered that the coefficients differ by a factor: βn = (d − 2n)!αn . In much of the following chapters I will suppress the indices by writing f (ψ) ≡ ψ a1 ...ad ǫa1 ...ad for ψ a d-form. For example: f (Ωn E d−2n ) = Ωa1 ...a2n ∧ E a2n+1 ...ad ǫa1 ...a2n . The wedge notation has also been suppressed.

(2.15)


22

2.5 Dimensionally continued Euler densities As mentioned earlier, the Lovelock action is closely related to the Euler Characteristic. The Euler Characteristic Class is reviewed in the appendix. The important relation for us is the following: Let g be a metric on a 2n dimensional manifold, M, of a certain topology. Let g ′ be another metric over M. Let ω, ω ′ be the torsion free, metric compatible, connection 1-forms associated with g and g ′ respectively. Then Z 1 n ′ n f (Ω(ω) ) − f (Ω(ω ) ) = nd dt f (θ Ω(t)n ). (2.16) 0

t is some parameter interpolating between the 1-form fields ω(t) ≡ ω + tθ, θ ≡ ω − ω ′, Ω(t) = dω(t) + ω(t) ∧ ω(t). The difference is just a total derivative. The proof involves the introduction of the homotopy parameter t. We shall make use of this and a natural generalisation to study intersections in later chapters. The Lovelock action (2.12) is a sum of terms very similar to the Euler density. There is just that extra factor of E (d−2n) . There are many nice features inherited from the Euler density. In particular, the variation of the connection, when restricted to infinitesimal variation, is still a total derivative, as we saw in equation (2.14).

3. THE JUNCTION CONDITIONS The action principle for General Relativity on a space-time manifold can be generalised to a manifold with boundary. The inclusion of a certain boundary term (Gibbons-Hawking) makes the action principle well defined on the boundary. Also, singular hyper-surfaces of matter [45] can be incorporated into a suitably smooth space-time. We will see that these are part of the general properties of actions built out of dimensionally continued topological invariants.

3.1 Hypersurfaces I will use the following definitions throughout: Definition 3.1. M is a d-dimensional manifold (usually assumed to have a Lorentzian metric). A hypersurface, Σ, is an embedded d − 1 dimensional submanifold. The bulk is the compliment of the hypersurface in M, M − Σ. Hypersurfaces typically divide M into disconnected bulk regions. Let us think of a simple situation. We have an oriented hypersurface Σ in Euclidean (or Minkowski) space. The position of the hypersurface is defined by the constraint: f (x) = 0. There is a normal vector at each point on Σ. The normal vector can be thought of as a vector in the Euclidean space. In other words it is assumed to live in the same vector space as the position vectors. In General Relativity space-time is intrinsically curved, with the curvature determined by matter content. as mentioned in the previous chapter, this means we have to do away with rigid Minkowski space in favour of a manifold. Not only do the positions now live in a more exotic object- a manifold, but the tangent vectors now live on the tangent bundle. In particular, the normal vector of the hypersurface lives on the tangent bundle. The non-Euclidean nature of the geometry makes the study of hypersurfaces more complicated. One useful tool is the Riemann normal co-ordinate system. On a local co-ordinate neighbourhood, we can always choose co-ordinates such that a smooth hypersurface is located at xd = 0 [49, 61]. It is not always convenient to work with this Riemann normal co-ordinate system. Sometimes the bulk solution will have a much simpler form in another co-ordinate system. Alternatively, the adapted frames introduced in the previous chapter may be useful. We have a hypersurface Σ, which can be defined by φ(x) = 0. In a surrounding region D we define a vector field n which coincide with normals on Σ. nµ ∝ ∂µ φ Using the Riemann normal co-ordinates mentioned above, the integral curves will be parameterised by some normal co-ordinate w(= φ).

3. The junction conditions

24

O n

integral curves of n

Fig. 3.1: D is a region of space-time around the hypersurface, Σ.

We define the induced metric (for a non-null hypersurface): hab = gab −

na nb n·n

(3.1)

and we will assume n is normalised to n·n = ±1 according to whether the hypersurface is time-like or space-like. In terms of n and h, the extrinsic curvature is: Kab = −hca ∇c nb

(3.2)

The details are in Appendix A.1. The extrinsic curvature tells us how the hypersurface is embedded into the bulk manifold. In flat space-time, this is purely determined by the intrinsic geometry of Σ. In non-Euclidean geometry there is the interesting possibility that the extrinsic curvature can actually be different on each side. The theory of singular hypersurfaces was introduced by Israel [45]. The Energy momentum tensor is assumed to be of the form: ab TBulk + T˜ ab δ(x ∈ Σ)

so that T˜ ab is singular with respect to the normal direction. The Israel Junction condition relates the energy-momentum of the wall to the discontinuity or jump of the extrinsic curvature: Σ T˜ ab = −[K ab − Khab ] ≡ −(K ab − Khab )|Σ+− ,

(3.3)


25

Σ± being the right or left hand side respectively. This is the singular part of the tangential-tangential component of the Einstein equation. The normal-normal and normal-tangential components of S ab must vanish. As discussed in chapter 2, the matter content is related to the geometry. In ab the thin shell formalism, with TBulk = 0, this is realised in a simple way. Since the Einstein Tensor is only non-zero at the location of the hypersurface, it merely determines the way in which the geometry of the two sides are to be matched together. It is not, however, as simple as matching flat regions. One has to match vacuum regions where the Weyl Tensor (the part of the curvature not appearing in Einstein’s equation) may not be zero. This part of the curvature depends, through the Bianchi Identity upon the global details of the matter-energy distribution such as the shape of the hypersurface itself. This formalism gives a well defined mathematical treatment of singular walls in GR. As an aside, I note that the initial value problem for hypersurface singularities was studied by Clarke [20] and by Vickers and Wilson [83]. There are some strange features of bubbles enclosed by singular shells in nonEuclidean space-time. For example, a bubble of de Sitter space inside a region of Schwarzschild space-time could be observed from just inside to be expanding while to an observer just outside it would be contracting [9]. The reason is that a wall of a given intrinsic shape can be embedded in a very different way into the half-manifolds on each side. We know that the effect of gravity is that, as we move along in space, the local inertial frames are relatively accelerated. If there is a smooth matter distribution this will be a continuous acceleration. If you step through a wall of singular mass distribution, there is a sudden jump in the acceleration. Imagine a man geodesically moving across such a wall. The geodesic is described by: x¨µ + Γµαβ x˙ α x˙ β = 0. The 4-velocity x, ˙ should be assumed continuous. The acceleration x¨ will be suddenly changed because the connection Γ is discontinuous. There would be a tidal force between the mans front leg on one side of the wall and rear leg on the other side. The above are curious properties of co-dimension 1 surfaces. When there is matter of co-dimension greater than one, a different geometrical feature becomes possible. If we parallel transport a vector in T (M) around some infinitesimal closed loop about our matter source, the final vector can be different from the initial. This type of curvature singularity with the localised holonomy is not possible with a co-dimension one brane since such a loop will always cut the brane once in each direction. However, such possibilities seem to lead to ambiguities in the non-linear theory of GR, as was shown by Geroch and Traschen [35]. The hypersurface is the only kind of singular source possessing a curvature tensor that is well defined as a unambiguous limit of a smooth concentration of matter.


26

3.2 Distributional curvature The curvature tensor involves second derivatives of the metric. Normally one would consider a metric that is at least C 2 (continuous, twice differentiable). This requires that the manifold is C 3 , i.e. the co-ordinate transformations between overlapping charts is at least C 3 . Then the curvature is a C 0 tensor. By Einstein’s Equations, this implies that the Energy-momentum Tensor is also C 0 . As mentioned in the introduction, Einstein himself felt that space-time should be smooth and conceived of fundamental particles as solitonic solutions of the field equations. We want to consider a less smooth case. We require only that the metric is piecewise C 2 . At the hypersurface it is only C 0 . The Riemann curvature tensor may then be singular. This requires that the differentiable structure of the manifold itself is piece-wise C 3 and C 1 at the hypersurface [19]. Taub [80] defined the distribution-valued curvature tensors in such a way that the Bianchi identity was still satisfied as an equation relating distributions. In GR, the curvature of a singular hypersurface is well defined as a distribution, as shown by Geroch and Traschen [35]. The singular solution, viewed as the limit of some non-singular matter is independent of the way that the limit is taken. This is in contrast to lower dimensional singular matter such as cosmic strings and point particles1 . However, the tensor product of distributions may not generally be a distribution. The field equations of higher order Lovelock theories involve the product of curvature tensors. It is not obvious whether this is well defined. Instead of addressing the problem of distributional field equations, for the Lovelock theory we shall just work with the Lagrangian. The boundary terms in the Lagrangian will define the equations of motion.

3.3 Gravity actions and boundary terms General relativity can be generalised to a manifold with boundary. The variational principle will however be a problem. The total derivative terms in the Euler-Lagrange variation will give a boundary term on ∂M. We would like to have the following variational principle: For fixed δhµν on the boundary2 , the action is to be minimised. Let us calculate the Euler-Lagrange variation of a general gravity action, keeping track of the total derivative terms. We have an action functional built by contracting products of the Riemann tensor with the metric tensor. S[g ab , ∂c g ab , ∂c ∂d g ab ] 1

It was shown by Garfinkle [32] that co-dimension 2 sources can be well defined in terms of distributional curvature, but they are not the unique limit of smooth matter. 2 Because of diffeomorphism invariance, this is the same as fixing δg µν on the boundary.


27

The Euler variation of the action with respect to the metric tensor is (see Appendix B.1): δS = H ab δg ab + ∂c V c ,

H

ab

∂L ∂L ≡ ab − ∂c ab + ∂d ∂c ∂g ∂g ,c

∂L − δg ab ∂d V ≡ δg ab ∂g ,c c

ab

∂L ∂g ab,cd

!

(3.4) ∂L ∂g ab,cd

!

+ ∂d δg ab

,

∂L . ∂g ab,cd

(3.5)

(3.6)

V c is the term which will appear on the boundary. If there were only terms proportional to δg µν and its derivative along the boundary, there would be no problem. The problem arises when there are normal derivatives of δg µν appearing on the boundary, coming from the last term in (3.6). These will not be fixed just by fixing δhµν on ∂M. In GR the inclusion of a certain boundary action (Gibbons-Hawking) cancels the normal derivatives of the metric variation. Thus the action can be extremised whilst keeping only the surface geometry fixed. This action was originally due to York [89] and was proposed within the context of the Hamiltonian formalism. It is also important for the path integral approach to quantum gravity [42]. In all of these cases, the boundary term which fixes the action is: Z p 2 n·n dd−1 x |h| K ∂M

The generalisation of this boundary term to Lovelock gravity was suggested by Myers [64]. It is simply related to the dimensional continuation of the Transgression associated with the Euler density (see appendix A.3). In terms of the moving frames, the general boundary term for the nth order Lovelock term is:3 Z Z 1 L(ω, ω0, e) ≡ n βn dt θa1 a2 ∧ Ω(t)a3 a4 ∧ · · · ∧ Ω(t)a2n−1 a2n ∧ ea1 ...a2n , (3.7) ∂M

0

where, as in Section 2.5, ω(t) ≡ ω0 + tθ, θ ≡ ω − ω0 , Ω(t) = dω(t) + ω(t) ∧ ω(t). It was argued by Myers [64] that the transgression formula (2.16) means that variation w.r.t. ω with ω ′ fixed just gives a closed form: δL(ω) = dδL(ω, ω0) 3

(3.8)

The expansion of this for n = 2 in terms of extrinsic and intrinsic curvature tensors is in [64] but there is a typo. The correct expression is in [22].


11111111111111111 00000000000000000 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 V1 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 1

28

111111111111111111 000000000000000000 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 V2 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 2

Fig. 3.2: The regions are glued together by identifying Σ1 with Σ2

in the vicinity of the boundary and that this relationship carries over to the dimensionally continued case. Hence, the combined action including the boundary term is one-and-a half order in the connection (for a fixed geometry on the boundary). Since the explicit variation of the action with respect to the vielbein can not produce any derivatives of a variation, we are guaranteed a well defined variational principle. This argument has been elucidated by Muller-Hoissen [63] and Verwimp [82]. Our approach will be somewhat different. The question that motivates the whole of what follows is: Can these boundary terms be used, and generalised, to describe membranes and intersections?

3.4 Gluing manifolds together A manifold with boundary is defined in the appendix. The boundary inherits its differential structure from the manifold. If f is some smooth tensor in the interior and can be extended smoothly across the boundary, then the derivative on the boundary is defined accordingly. We want to describe situations where matter is singular on hypersurfaces by an equivalent theory involving fields in the bulk and separate fields defined on hypersurfaces. I will treat the hypersurface as the shared boundary of two disconnected (open) regions V1 and V2 . In other words, the manifold is the union of the two regions and the hypersurface, M = V1 ∪ Σ ∪ V2 (fig. 3.2). For a well-defined embedding, this gluing process must also match up the tangent spaces in the correct way. It was shown by Clarke and Dray [19] that this gluing is well defined if the naturally induced metric on the hypersurface from each side are the same. The intrinsic geometry of Σ is independent of the embedding. There exists a unique C 1 (C 3 away from the hypersurface) co-ordinate atlas over M (giving


29

it a differentiable structure) which allows for all the components of the metric to be continuous across Σ. Define some (d-1)-form σ which is discontinuous at Σ and C 1 elsewhere. We want to integrate the exterior derivative of σ over M. In region V1 , σ coincides with some smooth (d − 1)-form field σ1 . Similarly, in V2 we have a smooth field σ2 . Because of the discontinuity at Σ : Z Z Z dσ2 . dσ1 + dσ 6= M

V2

V1

The inequality is due to the fact that dσ, evaluated on Σ, is not that induced by say σ1 continued across Σ. It is some kind of delta function. We can define Z Z Z dσ2 + IΣ , dσ1 + dσ = M

V2

V1

in such a way that Stokes’ Theorem is still valid on M. If M is without boundary: Z 0= i∗ (σ1 − σ2 ) + IΣ . Σ

we have used Stokes’ Theorem and ∂V1 = −∂V2 = Σ. Here i∗ σ1 means the pullback of σ1 with respect to the embedding i : Σ → M. I will usually neglect the i∗ notation and write, for example Z IΣ = (σ1 − σ2 ). Σ

So, integrating without this boundary term sees σ each side of the boundary as if it were to be smoothly continued across the boundary. In order to account properly for the derivative across the boundary, a boundary term must be added. In the Lovelock gravity action, when there is hyper-surface matter, we will encounter the exterior derivative of a discontinuous (d − 1)-form. We would like to represent it by an equivalent action involving fields in the bulk and separate fields with their support on hypersurfaces. The above calculations suggest that, if the divergent terms only appear in a total derivative, this is possible. It can be shown that Israel’s method for singular hyper-surfaces is equivalent to an action principle with boundary terms. In this case the variation of the metric evaluated on the hypersurface should not be fixed- it gives the junction conditions. The joining together of two boundaries with identical intrinsic geometry gives an action which yields the junction conditions. This has been done explicitly for Einstein’s theory by Hayward and Luoko [44] and the doubly covariant form of the action was shown by Mukohyama [62]. The doubly covariant action is where the intrinsic geometry is treated independently of the geometry of the bulk regions on either side. The equations of motion must, of course, generate the continuity of the geometry. I will now generalise to The Lovelock theory, giving a more simple proof where the bulk metric is varied and the intrinsic geometry is just treated as that induced


30

interior region D

1

2

Fig. 3.3: region D enclosed by two hypersurfaces.

from the bulk. I will show that the Lovelock action in the presence of singular hypersurfaces can be written in terms of smooth bulk integrals plus a boundary term. The question of the equivalence of the equations of motion is a more difficult one which I will go on to consider. This action for the n = 2 Gauss-Bonnet theory was first written down by Davis [22] and also independently by E. Gravanis and the author [36]. A general treatment of hypersurfaces in this theory has been done by Maeda and Torii [57]. Let ω be the Levi-Civita connection for the physical metric g. The manifold M admits also the metric b 2 = hij (xi , z = 0) dxi dxj + (n·n)(dz)2 ds

(3.9)

S = I1 + ID + I2

(3.10)

at least in some local co-ordinate neighbourhood, D, of Σ. Let ω0 be the connection associated with this metric. We are using explicitly Gaussian normal co-ordinates but θ ≡ ω − ω0 will be co-ordinate invariant. The one-form θ is the Second fundamental form of Σ. Following Mukohyama [62], we foliate the region D with hypersurfaces according to some congruence of differentiable curves (fig. 3.3). We will eventually take the limit D → Σ. We rewrite the action in terms of boundary terms which identically cancel4 :

4

for notation see (2.15)


I1 = αn I2 = αn ID = αn

Z

ZV1

ZV2 D

31

f (Ωn{1} E d−2n ) + IΣi , f (Ωn{2} E d−2n ) − IΣ2 , f (Ωn E d−2n ) + IΣ2 − IΣ1 .

There is a relative minus sign because the regions V1 and V2 induce opposite orientations on the wall Σ. That is to say the boundary of region V1 is ∂V1 = Σ + ... while the boundary of V2 is ∂V2 = −Σ + .... The boundary terms are chosen to be those already discussed (eqn. 3.7): IΣi ≡ n αn

Z Z Σi

1

dt f (θ{i} Ω(t)n−1 E d−2n ).

(3.11)

0

Let us show that there are no second normal derivatives appearing in ID . Our approach is somewhat simpler than that of Muller-Hoissen [63]. It will serve as an introduction to the formalism we shall use in later chapters to find the intersection terms. We interpolate between the connection ω and the connection ω0 associated with the induced metric. 5 ω(t) ≡ ω0 + tθ,

θ ≡ ω − ω0

Making use of several relevant identities in the Appendix A.3.1: Z 1 ∂ n d−2n n d−2n (3.12) f (Ω E ) − f (Ω0 E )= dt f (Ω(t)n E d−2n ) ∂t 0 Z 1 =n dt f (D(t)θ Ω(t)n−1 E d−2n ) 0 Z 1 Z 1 n−1 d−2n = nd dt f (θ Ω(t) E ) + n(d − 2n) dt f (θ Ω(t)n−1 D(t)E E d−2n−1 ). 0

0

The formula (2.16) for the Euler density becomes (3.12) for the dimensionally continued theory. The difference is the appearance of the last term: Z 1 Z ≡ n(d − 2n) dt f (θ Ω(t)n−1 D(t)E E d−2n−1 ). (3.13) 0

The total derivative term gives a contribution to the boundary, which is precisely minus our boundary term. So we can rewrite ID : Z Z 1 n d−2n n−1 d−2n−1 ID = αn f (Ω0 E ) + n(d − 2n) . (3.14) dt f θ Ω D(t)E E D

5

0

Note, we will not interpolate between E and E0 . We will do this in chapter (5)


32

This last term, Z, contains no second normal derivatives of the metric. This is because θ has a normal component in the local frame. Contraction with the antisymmetric Levi-Civita symbol means that only components of Ω(t) with tangential indices contribute. It is shown in the Appendix B.4 that these curvature components contain no second normal derivatives. We require that even in the limit D → Σ the metric is continuous. So the absence of second normal derivatives of the metric guarantee that ID is finite. We now take the limit D → Σ and ID must vanish, justifying the original definition. This leaves us with the bulk integrals on V1 and V2 and the boundary terms which become integrals on Σ± . Z Z Z S = L(ω1 , E) + L(ω2 , E) + L(ω2, ω0 , E) − L(ω1 , ω0 , E), (3.15) 1

2

Σ

where L(ω, E) ≡ L(ωi, ω0 , E) ≡

X

αn f (Ωn E d−2n ),

n

X n

αn

Z

1

dt f (θ{i} Ω(t)n−1 E d−2n ).

0

As an explicit case we give the argument due to Mukohyama for the Einstein theory [62]. Z Z Z p p d √ d−1 ID = d x −g (R − 2Λ) + 2 (n·n) d x |h|K − 2 (n·n) dd−1 x |h|K D

∂D+

∂D−

By a Gauss-Codacci decomposition [85], equivalent to equation (B.20), and partial integration, the boundary term is cancelled: Z √ ID = dd x −g d−1 R + (n·n)K 2 − (n·n)T r(K ·K) − 2Λ (3.16) D

As anticipated, there are no derivatives of K appearing in (3.16), so ID vanishes as D → Σ. This leaves us with the action: Z Z Z p d √ d √ S = d x −g (R − 2Λ) + d x −g (R − 2Λ) + (n·n) dd−1 x |h|(K1 − K2 ) 1

2

Σ

So we see that the Lagrangian (3.15) is well-defined as the singular limit of an integral over a smooth geometry. What does this mean for the field equations? I will illustrate the Einstein case, showing that the boundary method does indeed reproduce the Israel Junction conditions. The second fundamental form is, in terms of the extrinsic curvature tensor (B.13): θab = 2(n · n)n[a K b]c E c .


33

The boundary term is: α1

Z

Σ

(ω2 − ω1 )ab eab .

(ω2 − ω1 )ab = (θ2 − θ1 )ab = 2(n · n)n[a (K{2} − K{1} )b]c E c . We make use of the formula (A.22). L(ω1 , ω2 , E) = 2α1 (n · n)na (K{2} − K{1} )bc (δbc ea − δac eb ) = 2α1 (K{2} − K{1} )(n · n)na ea , where in the last line K ≡ Kaa . Also na Kab = 0 was used. So the hypersurface term is just the difference between the trace of the intrinsic curvature on each side, multiplied by the (d-1)-dimensional volume element (A.25): p (n · n)na ea = (n · n) |h| dd−1 ζ = e˜

in terms of the local co-ordinates on Σ. The surface term is the difference between the two Gibbons-Hawking boundary terms of regions 1 and 2. As shown in the next chapter, the infinitesimal variation of L(ω1 , ω2 , E) with respect to ωi gives two terms. One is a total derivative. The other cancels with the total derivative term coming from the bulk. The surface contribution to the equations of motion is just given by the variation with respect to the frame: δE C ∧ (ω2 − ω1 )ab ∧ eabc . The term multiplying δE is: 2(n · n)na (K{2} − K{1} )bd E d ∧ eabc = 2(n · n)na (K2 − K1 )bd (δad ebc − δbd eac + δcd eab ) = 2eãb (K{2} − K{1} )bc − (K{2} − K{1} )hbc

As found in (B.12) this equals −2T bc eãb . Equating the two terms, this is the Israel Junction condition Tãb = −β1 (K{2} − K{1} )ab − (K{2} − K{1} )hab . There are two approaches one can take to singular hypersurfaces: 1) Including boundary terms to give a well defined action principle. 2) Solving the equations of motion in the limit that sources become singular. Here, we have adopted the former approach. For Einstein’s theory, we have shown that the boundary approach reproduces Israel’s method, which is the unique singular limit of a smooth source. Methods 1 and 2 are equivalent.


34

For the higher order Lovelock Theory, we also have a well defined action principle with boundary terms. Consider some thick matter distribution where the width is taken to zero. Is integrating the field equations in the singular limit equivalent to our variational principle with a singular Lagrangian? This has been discussed by Barcelo et al [4] and by Deruelle and Germani [27]. It has been shown that the singular Lagrangian is equivalent to a solution of the field equations with distributional sources. What is not clear is whether this is a unique limit of smooth sources. If we want to approximate a dense but finite thickness shell of matter, such as domain walls or in simple models of black hole formation, method 2 would be appropriate. If we think of a membrane as some kind of fundamental particle, with matter strictly localised, then the singular Lagrangian approach, being a preferred way of describing the distributional field equations, seems to be more appropriate. Conjecture 3.2. For Lovelock gravity, methods 1 and 2 are equivalent. A proof of this would mean that our boundary approach is always a good approximation on scales larger than the thickness of the hypersurface. A general proof is not known to me.

4. INTERSECTIONS As mentioned in the introduction, much work has been done on the study of single and two brane worlds. Also there has been much work on the dynamics of a single false vacuum bubble in the context of inflation. We want to look at what happens in a more general network of hypersurfaces. It is well known from the obvious example of soap bubbles that when spherical bubbles come together they do not tend to remain as spheres. The surface energy of the bubbles tends to make them form into something like a honeycomb. There are shared boundaries and corners and edges. There have been several studies of collisions of shells of matter in General Relativity [28, 29, 51, 7]. Langlois et al [51] found an interesting geometrical treatment of colliding shells in AdS black hole backgrounds. We address the problem of intersecting/colliding walls in a more generalised way, motivated by the topological invariant approach presented in section 2.5. Our treatment will be for all Lovelock theories of gravity. We consider a smooth1 manifold with embedded arbitrarily intersecting hypersurfaces of singular matter. As in section 3.4, we can view a hypersurface as the shared boundary of two adjacent regions. The gravity of localised matter can be described by a boundary term in the action. We allow for the possibility of matter being localised on the surfaces of intersection also. More generally, the space-time is divided up into polygonal regions bounded by piecewise smooth hypersurfaces-like a matrix of bubbles or a honeycomb. We exploit the topological nature of the theory to write the action in terms of different connections in different regions. This generates our surface actions remarkably simply. We can derive the Junction conditions and the junction conditions for any intersections purely from the explicit Euler variation w.r.t the vielbein. To illustrate the problem we can think of a 3-way junction on the plane. The intrinsic geometry is well defined but the extrinsic curvature is discontinuous. At the 2-way junctions, although the metric is not differentiable, we can approximate differentiation in an unambiguous way. The three way junction is more problematic. There is no second derivative of the metric and no obvious way in which it can be well defined under integration. Our approach will be effectively to treat the exterior derivative as a certain kind of co-boundary operator in co-homology theory. In this way, the derivatives in the bulk contribute only to the walls and not the intersection. Similarly, the total derivatives on the walls contribute only to the co-dimension 2 intersections. The crucial fact is that the Lovelock Lagrangian is the dimensional continuation 1

C 1 : see section 3.2. This will amount to excluding any deficit angle at intersections.

4. Intersections

36

of the Euler density. The integral of the Euler density is a topological number and is independent of the discontinuities. We can dimensionally continue certain results over to the Lovelock theory to get a well defined unambiguous way of dealing with the lack of a second derivative.

4.1 Euler densities As mentioned in chapter 2, the Lovelock gravity is closely related to the Euler density. We shall first of all consider a ‘Lagrangian’ which is proportional to the Euler density. Such a theory is topological in the sense that it has no local degrees of freedom. The equations of motion are identically zero. We need to treat manifolds where the metric is continuous but not differentiable (C 0 ), the manifold being only once differentiable (C 1 ) [19]. The integral of the Euler density over a compact Riemannian manifold is equal to the Euler number. The Euler number is even defined for non-smooth objects such as polyhedra. This gives meaning to the integral over a manifold even when it is not smooth. This gives hope that we might be able to do something similar with a Lorentzian manifold and a dimensionally continued Euler density. The Euler density obeys the Transgression formula (2.16). Z 1 n n f (Ω1 ) − f (Ω2 ) = n d dt f (ω2 − ω1 ) Ω(t)n . 0

Let ω be a metric connection which has a discontinuity at some hyper-surface {12}. In region 1, the connection coincides with the smooth field ω1 and likewise region 2. Following the argument of section 3.4, we can write: Z Z Z Z L(ω) = L(ω1 ) + L(ω2 ) + L(ω1, ω2 ), M

1

2

{12}

L(ω) = f (Ω), Z 1 L(ω1 , ω2 ) = n dt f (ω2 − ω1 ) Ω(t) .

(4.1)

0

If M is Riemannian and compact, the above integral, with the hypersurface term, does indeed give the Euler number of M. When there exists an everywhere vanishing connection, the term L(ω, 0) is a Chern-Simons term. In that case L(ω) is an exact form: L(ω) = dL(ω, 0). Sometimes the more general term L(ω1 , ω2 ) is known also as a Chern-Simons term. If there are many regions separated by walls of discontinuity in the connection, the walls will intersect. We exploit the topological nature of the theory to write the action in terms of different connections in different regions. Integrating L(ω) over the manifold, when ω is the discontinuous connection form, one has to add the surface terms (4.1) integrated over the walls for the final result to have well defined variations with respect to ω (and to be diffeomorphism invariant). One should also add appropriate generalizations of the surface terms integrated over the intersection. In the next subsection we find these terms.

4. Intersections

37

4.1.1 From boundary to intersection action terms Given an Euler density the Transgression formula can be found by interpolating smoothly between the given connection ω and another one ω ′ . We can continue by interpolating between the latter and a new connection. In general, let us define the p-parameter family of connections, interpolating between p + 1 connections, ω1 , .., ωp+1, ω(p) ≡ ω(t1 ..tp ) ≡ ω1 − (1 − t1 )Υ1 − ... − (1 − t1 )..(1 − tp )Υp

(4.2)

where Υr ≡ ωr − ωr+1 ,

r = 1, .., p

Define p

X ∂ ∂ 1 p (1 − t1 )..(1\ − tq )...(1 − tr )Υr = Υq (t1 ...tbq ...tp ) ≡ Υq(p) ω = ω(t ..t ) = (p) q q ∂t ∂t r≥q

where the hat over a term or index means it is omitted. Let Ω(p) be the p-parameter curvature 2-form associated with ω(p) . Then ∂ Ω(p) =dΥq(p) + Υq(p) ∧ ω(p) + ω(p) ∧ Υq(p) ∂tq =D(p) Υq(p)

(4.3)

There is a generalised Bianchi identity for Ω(p) : D(p) Ω(p) = 0 (4.4) P P One way to prove this is to note that ω(p) = i C i ωi where i C i = 1 , see (B.28). Lemma 4.1. The generalisation of the term (4.1) to p + 1 curvature entries is: Z 1 L(ω1 , .., ωp+1) = ζp dt1 ..dtp f Υ1(p) Υ2(p) · · · Υp(p) (Ω(p) )n−p (4.5) 0

where

ζp = (−1)p(p+1)/2

n! (n − p)!

(4.6)

Proof: The proof is by induction. If we define ωp+1(tp+1 ) ≡ tp+1 ωp+1 + (1 − tp+1 )ωp+2 then: ωp+1 → ωp+1 (tp+1 ) ⇒ Υq(p) → Υq(p+1) ,

∀q ≤ p

4. Intersections

p+1

L ω0 , .., ωp (t

) = ζp

Z

38

1 0

dt1 ..dtp f (Υ1(p+1) · · · Υp(p+1) (Ω(p+1) )n−p )

We can write the difference of two terms by using this interpolation: Z 1 ∂ L(ω1 , .., ωp , ωp+1) − L(ω1 , .., ωp , ωp+2) = dtp+1 p+1 L ω1 , .., ωp , ωp+1(tp+1 ) ∂t 0 Z 1 ∂ (4.7) = ζp dt1 ..dtp dtp+1 p+1 f Υ1(p+1) Υ2(p+1) · · · Υp(p+1) (Ω(p+1) )n−p ∂t 0 From the multi-linearity of the invariant polynomial f we have ∂ n−p f Υ Υ · · · Υ (Ω ) = (4.8) 1(p+1) 2(p+1) p(p+1) (p+1) ∂tp+1 p X ∂ f Υ1(p+1) · · · p+1 Υr(p+1) · · · Υp(p+1) (Ω(p+1) )n−p = ∂t r=1 ∂Ω(p+1) n−p−1 + (n − p)f Υ1(p+1) Υ2(p+1) · · · Υp(p+1) (Ω ) (p+1) ∂tp+1 Using (4.3) we can write the last term as (n − p)f Υ1(p+1) · · · Υp(p+1) D(p+1) Υp+1(p+1) (Ω(p+1) )n−p−1 (4.9)

Now the covariant derivative D(p+1) Υp+1(p+1) does not just give a total derivative term. There will be other terms of the form D(p+1) Υq(p+1) . Making use of the generalised Bianchi Identity (4.4) and the invariance of the polynomial: p+1 X r=1

(−1)r−1 f Υ1(p+1) · · · D(p+1) Υr(p+1) · · · Υp+1(p+1) (Ω(p+1) )n−p−1 = D(p+1) f Υ1(p+1) · · · Υp+1(p+1) (Ω(p+1) )n−p−1 = df Υ1(p+1) · · · Υp+1(p+1) (Ω(p+1) )n−p−1

(4.10)

There is a total derivative term plus some other terms. It is natural to guess that they come from other L terms. To prove that this is the case, first note that

ω(t1 ..tp )|tr =0 =ω1 − (1 − t1 )Υ1 − · · · − (1 − t1 ) · · · (1 − tr−1 )(Υr − Υr−1 ) − · · · =ω1 − (1 − t1 )Υ1 − · · · − (1 − t1 ) · · · (1 − tr−1 )(ωr−1 − ωr+1 ) − · · · =ω(t1 , .., tr−1 , tr+1 , .., tp ) (4.11)

where ω(t1 , ..., tr−1 , tr+1 , ..., tp ) is the connection interpolating between ω1 , ..., ωr−1 , ωr+1 , ..., ωp+1 just as in (4.2). It follows that other L’s can be written: Z 1 ∂ n−p ζp dt1 · · · dtp+1 r f Υ1(p+1) · · · Υ\ r(p+1) · · · Υp+1(p+1) (Ω(p+1) ) ∂t 0 Z 1 r n−p t =1 cr · · · dtp+1 f Υ1(p+1) · · · Υ\ = ζp dt1 · · · dt r(p+1) · · · Υp+1(p+1) (Ω(p+1) ) tr =0 0

= 0 − L(ω1 , ..., ωbr , ..., ωp+2 )

(4.12)

4. Intersections

39

Evaluating the t-derivatives we do indeed get D(p+1) Υr(p+1) terms as well as others: ∂ n−p f Υ1(p+1) · · · Υ\ = r(p+1) · · · Υp+1(p+1) (Ω(p+1) ) r ∂t X ∂Υq(p+1) n−p \ f ··· · · · Υ · · · (Ω ) + r(p+1) (p+1) r ∂t qr

f Υ1(p+1) · · · D(p+1) Υr(p+1) · · · Υp+1(p+1) (Ω(p+1) )n−p−1

(4.13)

Bringing all the terms together by combining (4.8), (4.9), (4.10) and (4.13): p+1 X

∂ n−p \ f Υ · · · Υ · · · Υ (Ω ) = 1(p+1) r(p+1) p+1(p+1) (p+1) ∂tr = (n − p)df Υ1(p+1) · · · Υp+1(p+1) (Ω(p+1) )n−p−1 + ( p+1 X X ∂Υq(p+1) n−p · · · Υ\ (−1)r−1 f ··· + r(p+1) · · · (Ω(p+1) ) r ∂t r=1 qr X

In the last two terms, if we change variables r ↔ s in the latter and using this identity ∂ r ∂ ∂ ∂ Υp+1 = s r ωp+1 = r Υsp+1 (4.15) s ∂t ∂t ∂t ∂t we see that the terms cancel. Now if we substitute (4.7)(4.12) for the LHS in (4.14) we get the result: p+2 X r=1

(−1)r L(ω1 , ..., ω br , ..., ωp+2 ) = (n − p) ζp df Υ1(p+1) · · · Υp+1(p+1) (Ω(p+1) )n−p−1

We will multiply through by a factor of (−1)p+1 for later convenience. By comparison with (4.5), with p → p + 1 we get p+2 X r=1

provided

(−1)p+r+1 L(ω1, ..., ω br , ..., ωp+2) = dL(ω1 , ..., ωp+2) ζp+1 = (n − p)ζp (−1)p+1 n! ⇒ ζp = (−1)p(p+1)/2 (n − p)!

We have proved (4.5) by induction.

(4.16)

4. Intersections

40

It is not hard to show that L is fully anti-symmetric in its entries, so we can write (4.16) in the form p+1 X s=1

L(ω 1 , .., ω s−1, ω ′ , ω s+1, .., ω p+1) = L(ω 1, .., ω p+1) + dL(ω 1, .., ω p+1, ω ′)

(4.17)

where ω ′ is arbitrary. 2 The polynomial f is anti-symmetric with respect to interchanging two Υ’s so we have f (.., Υr , .., Υr , .., Ωp , ..Ωp ) = 0 and we can write (4.5) explicitly in terms of Υr = ω r − ω r+1 , r = 1..p in the form Z 1 1 p+1 L(ω , .., ω ) = dt1 ..dtp ζˆp f (Υ1 , Υ2 , ..Υp , Ωp , .., Ωp ), (4.18) 0

p(p+1) ζˆp = (−1) 2

p−1 Y n! (1 − tr )p−r . (n − p)! r=1

(4.19)

Let us finally show that Lp ’s, constructed from Characteristic Classes, are invariant under local Lorentz transformations. The connections transform: r ω(g) = g −1 ω r g + g −1 dg

for all r = 1, .., p + 1, where g belongs to the adjoint representation of SO(d-1,1). Then, Υr(g) = g −1 Υg and Ω(t)(g) = g −1Ω(t)g, so p+1 1 L(ω(g) , .., ω(g) ) = L(ω 1, .., ω p+1).

(4.20)

In fact, one can derive (4.17) without reference to the invariant polynomial , by use of the Poincare lemma and the following observation (inspired by the form of (4.17)). If f (x1 , ..xn ) is an anti-symmetric function of n variables and Af (x1 , ..xp , xp+1 ) = f (x1 , ..xp ) −

p X

f (x1 , .., xi−1 , xp+1 , xi+1 , ..xp )

i=1

(antisymmetrising over n+1 variables) then AAf (x1 , .., xp+2 ) = 0. The proof is trivial. 2

In different contexts the boundary variation of Chern-Simons terms are important (e.g. Gravitational and gauge anomalies [1] and black hole thermodynamics [33]). Here L3 is the boundary action of three “intersecting” Chern-Simons theories and as such it is gauge invariant under local Lorentz transformations.

4. Intersections

41

We can now show (4.17) by induction assuming only that it is true for the p=0 case. Assume (4.17) for p=k-1 (let us use the symbol Lk (ω 1 ..ω k ) for the intersection forms in this proof) ALk (ω 1..ω k+1 ) = −dLk+1 (ω 1..ω k+1 ) then AdLk+1 (ω 1 ..ω k+2) = 0. Since d is a linear operator, Ad − dA = 0 and so we get dALk+1(ω 1 ..ω k+2) = 0. By Poincare’s lemma, we have that there exists an invariant form, Lk+2 , locally, such that ALk+1(ω 1 ..ω k+2) = −dLk+2(ω 1 ..ω k+2). Since d is linear, the right hand side must be anti-symmetric w.r.t. the k + 2 entries. This completes the induction. There is similarity between our composition rule and Stora-Zumino descent equations [87], occurring in the study of anomalies in non-abelian gauge theories. Anomalies arise in a quantum field theory when the quantum effective action is not invariant under a classical gauge symmetry. To be well defined, an anomaly must satisfy the Wess-Zumino consistency conditions. It is useful to introduce ghost fields and the BRST operator which mixes between particles and ghosts. One can define a generating functional such that the consistency conditions are equivalent to the BRST invariance of this functional. The BRST operator is nilpotent: ss = 0. RIt anti-commutes with the exterior derivative: ds + sd = 0. Taking a functional T rF 2, F being the field strength, one can use the above procedure to establish the Stora-Zumino descent equations. This allows one to find BRST invariant integrals of non-BRST invariant terms (Chern-Simons type terms). The similarity with our derivation of the composition rule is the existence in both cases of a nilpotent operator, A in our case and the fermionic BRST operator there, which commutes and anti-commutes respectively with the derivative operator d.

4.1.2 Manifolds with discontinuous connection 1-form We now construct the action functional of gravity on a manifold containing intersecting surfaces. It will also enable us to draw conclusions for the general dimensionally continued topological R density. If the functional M L is independent of the local form of the metric of the manifold M, then it can be evaluated using a continuous connection as well as a connection that is discontinuous at some hypersurfaces. EitherRway the result will be the same. We use this formal equivalence to give a meaning to M L(ω) when ω is discontinuous. Let us start with the case of a topological density L(ω0) (ω0 continuous) integrated over M which contains a single hypersurface. Label 1 and 2 the regions of M separated by the hypersurface {12}. Introduce two connections, ω 1 and ω 2 which are smooth in the regions 1 and 2 respectively. We now write Z Z Z L(ω0 ) = L(ω1 ) + dL(ω1 , ω0) + L(ω2 ) + dL(ω2, ω0 ) (4.21) M

1

2

4. Intersections

42

3

{31}

{23} {123}

1 2 {12}

Fig. 4.1: The simplicial intersection of co-dimension 2 (h=2). The totally antisymmetric symbol {123} specifies the intersection including the orientation

Label the surface, oriented with respect to region 1, with 12. (Formally Z Z Z Z L(ω0 ) = L(ω1 ) + L(ω2 ) + L(ω1 , ω0 ) − L(ω2 , ω0 ) M 1 2 12 Z Z Z = L(ω1 ) + L(ω2 ) + L(ω1 , ω2 ) + dL(ω1, ω2 , ω0 ) 1

2

R

12

=−

R

21

).

(4.22)

12

That is, for a smooth surface the r.h.s. is independent of ω0 . Consider now a sequence of co-dimension p = 1, 2, 3..h hyper-surfaces which are intersections of p + 1 = 2, 3..h + 1 bulk regions respectively. A co-dimension p hypersurface is labelled by i0 ..ip where i0 , .., ip are the labels of the bulk regions which intersect there. We call this configuration a simplicial intersection. We take the example h = 2 (fig.4.1), where the intersections are {12}, {13}, {23}, {123}. An exact form integrated over {12} will contribute at {123} the opposite that when integrated over {21}, that is, for the latter integration the intersection can be labelled by −123 = 213, if we assume anti-symmetry of the label. The arrows of positive orientations in fig.4.1 tell us that a fully anti-symmetric symbol {123} will adequately describe the orientations of the intersection 123. This is in contrast to the non-simplicial intersection (fig. 4.2). Definition 4.2 (for a simplicial intersection). {i0 ...ip } is the set i0 ∩ · · · ∩ ip where P ir is the closure of the open set ir (a bulk region). ir overlap such that ∂ir = hs=0,6=r ir ∩ is and ir ∩ is = ∅ for all s 6= r. By ∂(A ∩ B) = (∂A ∩ B) ∪ (A ∩ ∂B), for A, B open sets, we can write: X {i0 ...ip+1 }. (4.23) ∂{i0 ...ip } = ip+1

4. Intersections

43

3 {34}

4

{23}

I

2

{41}

{12}

1

Fig. 4.2: A non-simplicial intersection of co-dimension 2 (h=2). The intersection, including the orientation, would not be properly represented by the totally anti-symmetric symbol {1234}

Full anti-symmetry of the symbol {i0 ...ip } keeps track of the orientations properly in (4.23). As a check: X {i0 ...ip+1 ip+2 } = 0. ∂ 2 {i0 ...ip } = ip+1 ,ip+2

Definition 4.3. A simplicial lattice is a lattice with all intersections being simplicial intersections (fig. 4.3). Definition 4.4. localised curvature on a surface of co-dimension > 1 is a singularity in the Riemann tensor such that the parallel transport of a vector around an infinitesimal closed curve produces a finite change. This means that there will not be a well defined ortho-normal frame at the intersection (see the Outlook section for a further discussion of this). Lemma 4.5. [37] For a simplicially valent lattice, with no localised curvature on intersections of co-dimension > 1 , the contribution from each intersection {i1 ...ik } is Z L(ωi1 , ..., ωik ) (4.24) {i1 ...ik }

up to a boundary term on ∂M. Proof: Assume, for l < h, Z

Z Z l−1 X 1 1 X L(ωi1 ..ωik ) + L(ω0 ) = L(ωi1 ..ωil ) + dL(ωi1 ..ωil , ω0 ). k! i ..i {i1 ...ik } l! {i1 ...il} M k=1 1

k

4. Intersections

44

Fig. 4.3: A simplicially valent lattice in two dimensions. Each intersection is the meeting of three bulk regions.

We have already seen that this is true for l = 1 and l = 2. The exact form gives Z 1 X L(ωi1 ..ωil ω0 ) + (a term on ∂M). l! i ..i i {i1 ...ilil+1 } 1

l l+1

From the anti-symmetry of {i1 ..il il+1 } and of L we have 1 X l! i ..i 1

l+1

Z

l+1

{i1 ...il il+1 }

1 X L(ωi1 ..ωir−1 ω ′ωir+1 ..ωil+1 ) l + 1 r=1

Applying the composition rule (4.17) we get: Z

Z l X 1 X L(ωi1 ..ωik ) L(ω0 ) = k! i ..i {i1 ...ik } M k=1 1 k X Z 1 + L(ωi1 ..ωil+1 ) + dL(ωi1 ..ωil+1 , ω0 ) (l + 1)! i ..i {i1 ...il+1 } 1

l+1

The total derivative term on the highest co-dimension intersections (order h), can only contribute to ∂M. Note that apart from our composition formula we have used only Stokes’ theorem, which is valid on a topologically non-trivial manifold M assuming a partition of unity fi subordinated to a chosen covering3 . By (4.20) each of the terms appearing will be invariant w.r.t. the P structure group. So the last formula is valid over M understanding each L as i fi L. By induction we have proved the Lemma. 3

This is reviewed in appendix A.2.2 as well as in the textbooks [18, 84].

4. Intersections

45

We began with a smooth manifold with an Euler density action which is completely independent of the choice of ω0 . This gives only a topological invariant of the manifold and is entirely independent of any embedded hypersurfaces. The ωi ’s, as well as their number, are arbitrary also. So we see that we have constructed a theory for arbitrarily intersecting hypersurfaces which is a topological invariant. It is a trivial theory in that the gravity does not see the hypersurfaces even if they contain matter.

4.2 Dimensionally continued action Now we consider the dimensionally continued Euler density for intersecting hypersurfaces separating bulk regions counted by i. Inspired by Lemma 4.5, we postulate the action: Sg =

Z h X 1 X Lg (ωi1 , .., ωik , e) Lg (ωi , e) + k! {i ...i } i 1 k i ..i k=2

XZ i

1

(4.25)

k

We will show that this action is ‘one and a half order’ in the connection. We will need to revisit our derivation of the composition rule in section 4.1.1, this time interpolating between the different vielbein fields E i (x), where the index represents the region (the local Lorentz index being suppressed). Physically, we require the metric to be continuous at a surface Σ1...p+1 : i∗ E i = E which implies i∗ (ei ) = e. Here i∗ is the pullback of the embedding of Σ1...p+1 into M. Define the Lagrangian on the surface Σ1...p+1 to be: 1

L(ω , . . . , ω

p+1

, e) =

(e(p) )a1 ...a2n =

Z

1 0

n−p dt1 · · · dtp ζbp f (Υ1 · · · Υp Ω(p) e(p) ),

(4.26)

1 (E(p) )a2n+1 ∧ .. ∧ (E(p) )ad ǫa1 ...ad . (d − 2n)!

where E(p) = E1 − (1 − t1 )(E1 − E2 ) − ... − (1 − t1 )...(1 − tp )(E(p) − E(p+1) ) and ζp is given by (4.19). Following through the calculation of section (4.1.1) , we pick up extra terms, involving derivatives of E(p+1) , from using the Leibnitz Rule on f . ∂ n−p f (Υ1(p+1) · · · Υp(p+1) Ω(p+1) e(p+1) ) = ∂tp+1 p+1 X n−p ∂e(p+1) )+ f (Υ1(p+1) · · · Υ\ s(p+1) · · · Υp+1(p+1) Ω(p+1) ∂ts s=1

n−p−1 (n − p)f (Υ1(p+1) · · · Υp+1(p+1) Ω(p+1) D(p+1) e(p+1) ) + (...)

The (...) denote terms which appear just as in section (4.1.1). We will verify our assertion that the action is one-and-a-half order by infinitesimally varying the metric and connection in one region. We vary them as independent fields. Using tp+1 to

4. Intersections

46

interpolate between Ep+1 and Ep+1 + δEp+1 and the corresponding variation of ωp+1 : δL(ω1 , . . . , ωp+1 , e) =

Z

0

Ξ=

p Y i=1

i

(1 − t )

p X

1

dt1 · · · dtp+1 ζbp Ξ + (...),

n−p ∂e(p+1) ) f (Υ1(p+1) · · · Υ\ s(p+1) · · · δωp+1 Ω(p+1) ∂ts s=1

n−p − f (Υ1(p+1) · · · Υp(p+1) Ω(p+1)

(4.27)

(4.28)

∂e(p+1) ) ∂tp+1

p Y n−p−1 D(p+1) e(p+1) ). + (1 − ti )(n − p + 1)f (Υ1(p+1) · · · Υp(p+1) δωp+1Ω(p+1) i=1

The (...) denote terms which will cancel when intersections are taken into account, just as in the topological theory (provided that the metric is continuous). Above, we have made use of Υp+1(p+1) = −(1 − t1 ) · · · (1 − tP )δωp+1. We want to check that the terms in (4.28) involving δωp+1 vanish. Now E(p+1) = E1 − (1 − t1 )(E1 − E2 ) − · · · + (1 − t1 ) · · · (1 − tp+1 )δEp+1 . Making use of formula (A.21) ∂ ∂ (e(p+1) )a1 ...a2n = (E(p+1) )b ∧ (e(p+1) )a1 ...a2n b (4.29) ∂ts ∂ts p X (1 − t1 ) · · · (1\ − ts ) · · · (1 − ti )(Ei − Ei+1 )b ∧ (e(p) )a1 ...a2n b + O(δEp+1 ). = i=1

So we see the first term in (4.28) vanishes if i∗ (Ei+1 ) = i∗ (Ei ) for all i = 1, . . . , p + 1 i.e. the metric is continuous. Given this, we see that: i∗ (D(p+1) E(p+1) ) = i∗ dE(p+1) + ω(p+1) ∧ E(p+1) (4.30) = i∗ d E1 + t1 (E2 − E1 ) + · · · + t1 · · · tp+1 δEp+1 + ω1 + t1 Υ1 + · · · + t1 · · · tp δωp+1 ∧ (E + t1 · · · tp+1 δEp+1 ) ∗

= i D(ω1 )E1 +

p X i=1

t1 · · · ti (D(ωi+1 )Ei+1 − D(ωi )Ei ) + O(δωp+1 ) + O(δEp+1 ) .

The third term in (4.28) already contains a δωp+1 apart from the D(p+1) e(p+1) . Now D(p+1) e(p+1) is proportional to D(p+1) E(p+1) so to first order in δEp , this term vanishes if D(ωi )Ei = 0 for all i = 1...p.4 4

The general gravitation theory we consider is built as a sum of a desired set of dimensionally continued topological densities. The coefficients can be taken to be functions of scalar fields, making the theory dilatonic. The variation with respect to the connection leads to a non-zero torsion in this case. The torsion equation is a constraint on the variation of the scalar fields. Solving it for the connection and substituting in the Lagrangian one obtains explicitly an action for the dilatonic fields.

4. Intersections

47

The only non vanishing term in (4.28) is the second which involves: ∂ (e(p+1) )a1 ...a2n = − (1 − t1 ) · · · (1 − tp )(δEp+1)b ∧ (e(p+1) )a1 ...a2n b ∂tp+1 = − (δe(p) )a1 ...a2n . So we arrive at a simple expression for the variation of the action, once the equation of motion for the connection and continuity of the metric have been substituted: Z 1 δL(ω1 , . . . , ωp+1, e) = dt1 · · · dtp ζbp f (Υ1 · · · Υp Ωpn−p δe) + (...). 0

Then, variation of an ωi will vanish automatically upon imposing the zero torsion condition and the continuity of the metric at the intersections5 . Second, from the variation of the frame E a we obtain field equation for gravitation and its relation to the matter present, by δE Sg + δE Smatter = 0.

(4.31)

The field equations are actually algebraically obtained, on the gravity side, using (A.21). Note that although intersections describe physically situation such as collisions there is a non-zero energy momentum tensor at the intersection when the theory is not linear in the curvature 2-form. The dimensionally continued n-th Euler density produces a non-zero energy tensor down to d − n dimensional intersections. Explicitly, the gravitational equation of motion for a simplicial intersection {1, ..., p + 1}, carrying localised matter Lm(1...p+1) is p

(−1)

X n

αn

Z

0

1

δ dt1 · · · dtp ζˆp (Υ1 · ·Υp )a1 ...a2p (Ω(p) )a2p+1 ...a2n eba1 ...a2n = Lm(1...p+1). δE b (4.32)

We have dropped the wedge notation. The (−1)p factor comes from permuting δE b to the left hand side.

5

The equations of motion for ω are satisfied by the zero torsion condition but are not, for n ≥ 1 identical to it. There are potentially solutions of non-vanishing torsion [58].

5. HOMOTOPY PARAMETERS AND SIMPLICES It was shown in the previous chapter that one can formally rewrite the Euler number in terms of a discontinuous connection. One will then have boundary and intersection terms in the integral. This amounts to turning the manifold into a honeycomb-type lattice. The action, including boundary terms was found to be: Z XZ X 1 XZ L(ωi1 , ..., ωik ). (5.1) L(ω) = L(ωi ) + k! i ...i {i1 ...ik } M i i k≥2 1

k

which came from the composition rule p X s=1

(−1)s−p−1L(ω1 , .., ωbs , .., ωp ) = dL(ω1, .., ωp )

(5.2)

L(ω) = f (Ωn ) and explicit formulae for the intersection terms were found. We will find somewhat simpler expressions for them in the next section. I shall re-derive these results by introducing a closed form η in a space W ⊂ F . F is a product space of the manifold M and the space of Homotopy parameters appearing in the definition of the intersection terms. Our composition rule (5.2) will be shown to be equivalent to The condition that η be closed. The results can be presented in a simpler way by introducing a multi-parameter generalisation of the Cartan Homotopy Operator. The entire honeycomb is described by a few simple equations. All sorts of intersections are accommodated in the scheme given by these equations and by the shape of W . For the Euler density, we find an explicit expression1 for η and show that it is closed in F . The form of the intersection terms will be clarified greatly. Then we turn to a dimensionally continued action where the metric enters into the action. We will show that the gravitational intersection Lagrangians obey the same composition rule (5.2), further justifying our action (4.25). This is because the dimensionally continued η is still closed over the domain of integration W ⊂ F . We can write the action which generates all the intersection terms as: Z S= η, (5.3) W

where η is given by (5.8) for the Euler density and (5.28) for the dimensionally continued Euler density. 1

This form η is already known in the mathematics literature [31]. It is the Secondary Characteristic form. The application of this approach to the dimensionally continued Euler form is a new departure.

5. Homotopy parameters and simplices

(a)

(b)

49

(c)

Fig. 5.1: (a) Simplicial intersection; (b) Simplex in t-space; (c) A projection of W .

5.1 A geometrical approach We want to describe the situation in the vicinity of an intersection of co-dimension p between different bulk regions. In this vicinity there will also be intersections of lower co-dimension. At each intersection, we have a meeting of connections ωi in the different regions. Let us for the moment deal only with simplicial intersections. We define the simplicial intersection of codimension p to be a surface of codimension p where p + 1 regions meet (fig. 5.1a). It was found in chapter 4 that the L(ω0 , ..., ωp ) is an integral over p different homotopy parameters interpolating between the connections Let us interpolate in the most symmetrical way. We introduce a p-dimensional simplex (appendix A.2.3) in the space of some parameter t (fig. 5.1b ). Let us define the interpolating connection: ω(t) ≡

p X

i

C (t) ωi ,

p X

C i (t) = 1.

(5.4)

i=0

i=0

SometimesPwe will use the specific parameterisation denoted by the Latin index ti . C 0 = 1 − pj=1 tj and C i = ti , i = 1, .., p. ω(t) = ω0 +

p X i=1

ti χi ,

χi ≡ ωi − ω0 .

(5.5)

To avoid confusion we shall use a Greek index tα to denote the general parameterisation. Each order of intersection causes us to lose a dimension but gain an extra connection. each new connection means an extra parameter of continuous variation. As it were, in integrating, each time we lose a dx we gain a dt. With this in mind, we can think of our action as living in a d-dimensional space which is a mixture of space-time and t directions. So we introduce the Space F = M × SN −1 , with SN −1 , a simplex of dimension N − 1, N being the total number of regions on M. At each of the N points of the simplex there lies a continuous connection form ωi on M with its support on some


50

open set in M containing the region i. Each contribution to our action will live on some d-dimensional subspace of F . The technical reason for introducing this is that the connection is continuous on F and integration is well defined. There is also an aesthetic reason. It is quite a nice feature of the problem that the mathematics will take on its simplest form when the t-space is a simplex. It provides a geometrical picture which can simplify many calculations. For example, the treatment of a nonsimplicial intersection becomes easy as we shall see. Let us define a d-dimensional differential form in this space F (where for convenience the dx’s are suppressed). n X 1 α1 dt ∧ · · · ∧ dtαl ∧ ηα1 ...αl (x, t), η≡ l! l=0

(5.6)

ηα1 ...αl ≡ ηα1 ...αlµl+1 ...µd dxµ1 ∧ · · · ∧ dxµd .

We can now proceed to integrate this form over different faces of SN −1 . A p-face (which we call s0...p or just s) is a subsimplex of SN −1 which interpolates between a total of p + 1 different connections. Let us define L0...p to be the integral over the p-dimensional face:2 Z Z 1 dtα1 ∧ · · · ∧ dtαp ∧ ηα1 ...αp , (5.7) η= L0...p ≡ p! s0...p s0...p η here being understood to be evaluated at t = t(s) so that the integral is a function of x only. This integral picks out terms in η which are a volume element on the appropriate face. We would like, for an appropriate choice of η, to identify this term with L(ω0 , ..., ωp ) as previously defined with respect to the Euler density. We shall see that this can indeed be done and we shall find a simple form for η. Definition 5.1. d(F ) = d(M ) + d(t) is the exterior derivative on F . d(t) and d(M ) are the exterior derivative restricted to the simplex and M respectively. Proposition 5.2. The appropriate condition on η such that L0...p obeys the composition rule (5.2)is that η be a closed form, d(F ) η = 0. Proposition 5.3. The form of η corresponding to the Euler density is ∧n η = f d(t) ω(t) + Ω(t) .

(5.8)

Ω(t) is the curvature associated with ω(t). A similar form has previously been considered in ref. [31]. 2

Strictly there should be a factor of (−1)P (0,...,p) in the middle term to account for the orientation with respect to SN −1 . However, we can choose s to have the positive orientation by assuming the points 0...p are in the appropriate order.


51

Proposition 5.4. The intersection Lagrangian can be recovered by the specific choice: η1...p = Ap f χ1 ∧ ... ∧ χp ∧ Ω(t)∧(n−p) , (5.9) n! Ap = (−1)p(p−1)/2 (n − p)! Z dp tf χ1 ∧ ... ∧ χp ∧ Ω(t)∧(n−p) . (5.10) ⇒ L(ω0 , ..., ωp ) = Ap s01..p

P i χi ≡ ωi − ω and ω(t) ≡ ω + 0 0 α t ωi . The integration is over the right angled P simplex {t| i ti ≤ 1}. We will see that the asymmetry between the point 0 and the points 1, ..., m is merely an illusion.

To prove Proposition 5.2, we will need to use Stokes’ Theorem on the face, s. Z Z d(t) η = η. (5.11) s

∂s

The boundary of the simplex s0...p is ∂s0...p

p X (−1)i s0...bi...p = i=0

with the orientation being understood from the order of the indices. Now let us integrate the form d(M ) η over the face. We will need to remember that in permuting this exterior derivative past the dt’s we will pick up a ± factor. d(M ) η =

X (−1)l l!

l

dtα1 ∧ · · · ∧ dtαl ∧ d(M ) ηα1 ...αl .

Using this information we may integrate over the p-face s Z Z p d(M ) η = (−1) d η. s

(5.12)

s

Combining equations (5.11) and (5.12): Z

p

d(F ) η = (−1) d

Z

η+

s0...p

s0...p

p X i=0

i

(−1)

Z

η.

(5.13)

s0...bi...p

If our form η is closed in F, d(F ) η must necessarily vanish term by term in the dt’s and dx’s. The integral on the right hand side of (5.13) must therefore vanish. Recalling the definition (5.7) we have proved Proposition 5.2: d(F ) η = 0

⇔

dL0...p =

p X i=0

(−1)p−i−1 L0...bi...p

(5.14)


52

The condition that η be closed is indeed equivalent to our composition formula.

The proof of Proposition 5.3 is in Appendix B.5. Proposition 5.4 follows from proposition 5.3 by expanding the polynomialP but we will show it by more brute force method. First we note that for ω(t) = ω0 + i ti χi we get a useful formula: ∂Ω(t) = D(t)χi ∂ti

(5.15)

where D(t) is the covariant derivative associated with ω(t). Now let us verify explicitly that the right hand side of (5.13) vanishes. For convenience, we will drop the wedge notation. Z

∂s

p X

Z

∂ η (5.16) i 1...bi...p ∂t i=1 Z p X ∂Ω(t) i−1 (n−p) (−1) (n − p + 1)Ap−1 dt1 · · · dtp f χ1 · · · χbi · · · χp = Ω(t) ∂ti s i=1 Z p X i−1 1 p n−p (−1) (n − p + 1)Ap−1 dt · · · dt f χ1 · · · D(t)χi · · · χp Ω(t) =

η=

i−1

(−1)

dt1 · · · dtp

i=1

=(n − p + 1)Ap−1dM Ap−1 =(n − p + 1) dM Ap

s

Z

Z

dt1 · · · dtp f χ1 · · · χp Ω(t)n−p η

s

In the first line we have used Stokes’ Theorem (5.11). In the second and last line (5.10) was used. In the third we made use of (5.15). In the fourth the Bianchi identity for Ω(t), (B.28) and the invariance of the polynomial (B.30). A comparison with equation (5.13) tells us that dF η does indeed vanish provided Ap = (n − p + 1)(−1)p−1 Ap−1

⇒

Ap = (−1)p(p−1)/2

n! (n − p)!

There is a consistency check we need Pto make. We want to equate the left hand side of (5.16) with a sum of terms pi=0 (−1)i−1 L(ω0 , ..., ωbi, ..., ωp ) . (5.10) is not manifestly symmetrical since it is constructed on a right-angled simplex with ω0 at the origin. It follows straightforwardly from (5.9) for i 6= 0 that Z η = L(ω0 , ..., ωbi, ..., ωp ) (5.17) s0..î...p


53

What about the integral over the opposite face s1...p ? Z Z X i ci · · · dtp f (χ1 · · · χbi · · · χp Ω(t)n−p+1 ) dt1 · · · dt η= (−1) s1...p

i

i

= (−1)

XZ i

p−1

= (−1)

(5.18)

s1...p

Z

1

0

dt · ·

1 1

0

Z

1

dt · ·

1−

P p−2

tj

j=1

j=1

0

Z

1−

P

j

dtp−1 f χ1 · ·χbi · ·χl Ω(t)n−p+1 |tp =1−P p−1 tj p−1

tj

m−1

dt

f

0

^

(ωk − ωp ) Ω(ωp + Σi ti (ωi − ωp )n−p+1

k=1

= L(ω1 , ..., ωp )

The integral has been made manifestly equivalent to an integral over a right angled simplex with the origin at ωp . In the second line, we have made use of the fact that ci with dt cp . In the the multiple integrals are all over the same face to replace the dt third line, we have used: p

ω(t = 1 −

j Σp−1 j=1 t )

= ω0 +

p−1 X i=1

i

t χi + (1 −

p−1 X

i

t )χp = ωp +

p−1 X i=1

i=1

ti (χi − χp )

Also we have made use of the following relation, obtained from the multilinearity and anti-symmetry of the function f with respect to the χ’s. p−i−1

f (χ1 − χp ) · · · (χp−1 − χp ) · ··) = (−1)

p X i=1

f (χ1 · · · χbi · · · χp · ··).

This makes the anti-symmetry of L with respect to the connections clear. Combining (5.9), (5.17), (5.18), (5.16) and (5.14) completes the proof of Proposition 5.4.

5.2 The implications We have introduced a more abstract approach. What have we gained by this? 5.2.1 Dual simplices Firstly we see that the simplicial intersection is related to a simplex in the parameter space. It is a bit like turning the simplex inside out. As pointed out already, the connection is smooth on F = M × SN as the d-dimensional Lagrangian density η weaves its way through x and t space. We can check that the form of the intersection action is found in this chapter is the same as in the previous chapter, (4.18). The difference is that here the domain of integration is over a simplex (also Ω(t) is defined in a seemingly different way). The measure for the simplex is Z

1 1

dt 0

Z

1−t1 2

0

dt · · ·

Z

0

1−

P p−1 i=1

ti

dtp .


54

Let us make the co-ordinate transformation: ti → si =

1−

ti Pi−1

j j=1 t

.

Now the domain of integration is as in the previous chapter. We want to write the measure in terms of s. So we will need to find the inverse Jacobian matrix of this co-ordinate transformation. The Jacobian matrix is in upper diagonal form so the determinant is just the product of diagonal entries. p

p

∂si Y ∂si Y 1 det j = . = P j ∂t ∂ti 1 − i−1 j=1 t i=1 i=1

The following formula is easy to prove: 1−

i−1 X j=1

j

1

t = (1 − s ) · · · (1 − s

i−1

i−1 Y )= (1 − sj ).

(5.19)

j=1

The determinant of the inverse matrix is, making use of (5.19): p

p−1

i−1

Y YY ∂ti det j = (1 − si )p−i . (1 − sj ) = (1 − s1 )p−1 (1 − s2 )p−2 · · · (1 − sp−1 ) = ∂s i=1 i=1 j=1 So the measure is, in terms of the s parameters: Z

0

1

···

Z

p−1

1 1

0

p

dt · · · dt

Y (1 − si )p−i i=1

This is the same as the measure derived in chapter (4). The discrepancy in the minus sign factors between (5.9) and (4.19) is a factor of (−1)p . This comes from the fact that Υi = −χi . It just remains to prove that Ω(t) = Ω(p) This follows from ω(p) =ω0 + (1 − s1 )(ω1 − ω0 ) + · · · + (1 − s1 ) · · · (1 − sp )(ωp − ωp−1 ) p X 1 2 p tj )ωp = ω(t) =t ω0 + t ω1 + · · · + t ωp−1 + (1 − j=1

ωp is at the “corner” of the simplex not ω0 , but we have already seem that the integral is independent of the choice of the corner. We have confirmed that the two definitions of the intersection forms L(ω0 , . . . , ωp ) are equivalent. We have proved the same result by two independent approaches. 5.2.2 F-space and Homotopy Operator Secondly, we have a simple expression for the Lagrangian density in W ⊂ F . It is given by (5.8) (recall W is the region of integration in F ). From equation (B.27) we


55

notice that d(t) ω(t) + Ω is just a kind of curvature of ω(t) on F , call it ΩF (t). In other words, the action is very trivial on this enlarged space. Z S= η, η = f (ΩF (t)n ) (5.20) W

and it obeys the same transgression formula as the thing we started with, only now on F . Under continuous variation ω(t) → ω ′(t), f (ΩF (t)n ) − f (ΩF ′ (t)n ) = dF TP (ω(t), ω ′(t)). The shape of W is interesting. Every d − 1 dimensional surface is thickened in the t-direction by a 1-dimensional simplex; These meet at a d-2 surface in M which looks like a triangular prism in W , etc. (fig. 5.1(c)). The proof of proposition 5.2 can be thought of in terms of a generalisation of Cartan’s Homotopy formula to a Rhigher number of homotopy parameters. Let the R operator Ks be defined by Ks η ≡ s η and let K∂s ≡ ∂s η. The equation 5.13 can be written as (Ks · dF − (−1)m dF · Ks )η = K∂s η

(5.21)

This reduces to the usual Cartan Homotopy Formula for the 1-simplex m = 1. (K01 dF + dF K01 )η = η(1) − η(0) In fact, the whole of our analysis can be reduced down to the two equations (5.20) and (5.21) In words: The whole intersection Lagrangian is a density in some higher dimensional simplicial product space over our manifold. The composition rules are an expression of this higher dimensional Cartan Homotopy operator acting on this Density. 5.2.3 Non-simplicial intersections made simple Thirdly, we now have a very efficient way to deal with a non-simplicial intersection in M. This is where k > p regions meet at a co-dimension p surface. We can easily deal with a non-simplicial intersection by integrating over a simplicial complex in t-space. More than one face of dimension p are associated with the same (d − p)-surface in M. Lets consider a simple example. We have 4 regions, 1, .., 4, meeting at a codimension 2 intersection I ⊂ M (fig.5.2). There are four hypersurfaces {12}, {23}, {34} and {41} meeting at I. As described in Lemma 4.5, on each hypersurface lives a term Lij = L(ωi , ωj ). Integrating η over the simplicial complex s123 ∪ s341 will give us the intersection term. The complex has boundary s12 ∪ s23 ∪ s34 ∪ s41 . Applying (5.13) and d(F ) η = 0: Z Z η η =d s12 ∪s23 ∪s34 ∪s41

s123 ∪s234

⇒ L12 + L23 + L34 + L41 =d (L123 + L341 )

(5.22)


56 4

4

1

3 3 1 2 2

Fig. 5.2: Non-simplicial Intersection. (a) Space-time, (b)t-space.

The appropriate term for the non-simplicial intersection I is Z Z η I⊂M

s123 ∪s341

or equivalently we can integrate over s234 ∪ s412 which has the same boundary. More symmetrically, integrate over the chain c = 12 (s123 + s234 + s341 + s412 ). Z Z η. (5.23) I⊂M

c

So the term which lives on the intersection is 12 (L123 + L234 + L341 + L412 ). It is the degenerate case where two simplicial intersections {123} and {341} (or equivalently {234} and {341}) coincide.


57

5.3 Dimensionally continued Euler density So far we have been considering the topological density. This is not suitable as a Lagrangian. We know that the action yields no equations of motion. The point is that we can apply what we have learned to the dimensionally continued densities. The Lovelock Lagrangian is a combination of such densities: [d/2]

L=

X

αn f (Ωn E d−2n ),

(5.24)

n=0

f (Ωn E d−2n ) = Ωa1 a2 ∧ · · · ∧ Ωa2n−1 a2n ∧ E a2n+1 ∧ · · · ∧ E ad ǫa1 ...ad . We assume that the connection is a metric compatible (Lorentz) connection. There are now explicit factors of the vielbein frame E a appearing in the action. We have a manifold M, of dimension d, with regions, i, divided by walls of matter. The metric on M is continuous but the derivative of the metric is discontinuous at the surfaces. Once again we rewrite the Lagrangian in terms of the continuous connections ωi and boundary terms. We interpolate as before: E(t) ≡

p+1 X

ω(t) ≡

Ci (t) Ei ,

i=1

p+1 X

Ci (t) ωi ,

p+1 X

Ci (t) = 1.

(5.25)

i=1

i=1

The quantities D(t)E(t) and d(t) E(t) vanish everywhere on our domain of integration W . In fact, a good way to define W is: W is the region in F where E(t, x) = E(x). (of course d(M ) E(t, x) is a function of t because the derivative of the metric is discontinuous on M). (x, t) ∈ W,

E(x, t) = E(x), D(t)E(t)a =

X

C i d(M ) Eia +

i

=

X i

X i

C

i

ωi ab

(5.26)

C i ωi ab ∧ E(t)b

∧ (E(t) − Ei )b = 0,

(5.27)

(x, t) ∈ W.

We have used the zero torsion condition d(M ) Eia + ω ab ∧ Eib = 0. The case of a single hypersurface is illustrated in figures 5.3 and 5.4. At the intersection, E 1 = E 2 . At the intersection and only at the intersection, both are fields living on W . D(t)E(t) = 0, where E(t) = C(t)E 1 + (1 − C(t))E 2 . In region 1, the above formula does not hold, we only have D 1 E 1 = 0. Proposition 5.5. The composition rule, (5.2), holds for the Lovelock action, when evaluated at a point on M where all the regions 1, ..., p intersect. Define the form: ηDC = f (d(t) ω(t) + Ω(t))∧n ∧ E(t)∧(d−2n) (5.28)

The terms entering into the composition rule will be terms in the expansion of ηDC picked out by integrating over the appropriate simplex s1...p in F .


58

E1(x) E(x,t) E2(x) region 1

{12}

x

region 2

Fig. 5.3: E(x,t)=E(x) at the intersection {12}

2

(t)

1

region 1

{12}

region 2

Fig. 5.4: ω(t) interpolates between the ωi .

x


59

We need just to show that ηDC is closed when restricted to the space W . The proof then follows from Proposition 5.2. We make use of the invariance property of the polynomial contracted with the epsilon tensor. In W : d(F ) f (d(t) ω(t) + Ω(t))∧n ∧ E(t)∧(d−2n) = DF (t)f (d(t) ω(t) + Ω(t))∧n ∧ E(t)∧(d−2n) =0 We have defined the covariant derivative on F : DF (t) ≡ d(t) + D(t). The integral vanishes because DF (t)(d(t) ω(t) + Ω(t)) = 0 by (B.31) and DF (t)E(t) = 0 as explained above. For the dimensionally continued case, ηDC and L are no longer Euler densities. It was therefore not obvious that our composition formula should survive. It does survive though because ηDC is still a closed form in W ⊂ F . As a consequence of the composition rule, we can apply exactly the same argument as in Lemma 4.5 to prove: Lemma 5.6. The Lovelock action, in the presence of hypersurfaces, is given by (4.25) which we may now write as Z S= ηDC (5.29) W

Also, as we saw in the previous chapter, the infinitesimal variation of the action with respect to the connection vanishes, provided we impose the torsion free condition on the connection and continuity of the metric. So the equations of motion just come from the explicit variation with respect to the vielbein.


60

5.4 The interpolating curvature We find the explicit form of the interpolating curvature:3 Ω(t) =

X i

X

C i dωi +

1X i j C C [ωi , ωj ] 2 i,j

1 C i C j dωi + [ωi , ωj ] 2 i,j 1 1 1X i j C C Ωi + Ωj − [ωi , ωi ] − [ωj , ωj ] + [ωi , ωj ] = 2 i,j 2 2 X = C i C j Aij , =

i,j

Aij =

1 1 Ωi + Ωj − [ωi − ωj , ωi − ωj ] . 2 2

Making use of Appendix B.3, 1X i j X1 Ω(t) ∼ CC Ω(ik) − n·n(ik) K(i) ∧ K(i) 2 i6=j p k6=i X1 Ω(kj) − n·n(kj)K(j) ∧ K(j) + p k6=j + n·n(ij) K(i) ∧ K(i) + K(j) ∧ K(j) − 2K(i) ∧ K(j) where the symbol ∼ means excluding terms with normal Lorentz indices.

(5.31)

(5.32)

5.5 Intersecting hypersurfaces in more general theories via closed forms The existence of the closed form η is very important for the description of intersecting membranes. The question arises: are there any other forms which can be generalised to closed forms in W space? We have dealt with Lovelock Lagrangian because it includes the Einstein-Hilbert Lagrangian of General Relativity, and because of the many nice mathematical and physical features discussed in Chapters 1 and 2. The relationship between the Lovelock Lagrangian and the Euler Characteristic Class was important in the construction 3

We can see from(5.31) that even when the bulk regions themselves are flat, the interpolating curvature may not vanish Ωi = 0, ∀i X 1 C i C j [ωi − ωj , ωi − ωj ]. ⇒ Ω(t) = 2 i,j

(5.30)


61

of the closed form. In looking for more generalised forms, the first thing to consider are other Characteristic Classes. A general Characteristic form, P (Ωn ) can be generalised to the Secondary Characteristic form [31] P ([dt ω(t) + Ω(t)]n ).

(5.33)

For example, in four dimensions, there is the Pontryagin form: (the Hirtzebruch signature) τ ∝ Ωab ∧ Ωba .

(5.34)

In this case, the invariant tensor which contracts the indices is ηa2 a3 ηa4 a1 rather than ǫa1 ...a4 . The more familiar tensor form of the Hirtzebruch signature is: √ ∝ gRµναβ Rρσγδ ǫαβγδ gµρ gνσ . The Secondary Characteristic form associated with the Hirtzebruch signature is: τ = [dt ω(t) + Ω(t)]ab ∧ [dt ω(t) + Ω(t)]ba

(5.35)

which is clearly a closed form in F , (d(t) + d(M ) )τ = 0. In higher dimensions, there are other Pontryagin forms. for example, in eight dimensions, there are two such forms: Ωab ∧ Ωbc ∧ Ωcd ∧ Ωda ,

(Ωab ∧ Ωba ) ∧ (Ωcd ∧ Ωdc ). These forms characterise the topology of M (or of the fiber bundle if Ω is the curvature of a gauge connection). To describe a theory with local degrees of freedom, we need to generalise them. There are various ways of doing this, but all of them involve a non-zero torsion. The forms that one can make are [58]: RA := Ωaa12 ∧ · · · ∧ ΩaaA1 ,

(5.36)

VA := Ea1 ∧ Ωaa12 ∧ · · · ∧ ΩabA ∧ E b , TA := Ta1 ∧ Ωaa12 ∧ · · · ∧

KA := Ta1 ∧

Ωaa12

∧ ···∧

ΩabA ΩabA

∧ T b, b

∧E .

(5.37) (5.38) (5.39)

There are various ways of forming d-forms out of products of these. ζ = RA1 ∧ · · · ∧ RAr ∧ TB1 ∧ · · · ∧ TBt ∧ VC1 ∧ · · · ∧ VCv ∧ KD1 ∧ · · · ∧ KDk , (5.40) where 2(

r X i=1

Ai +

t X

Bi +

i=1

Ai , Bi even,

v X i=1

Ci +

k X

Di ) + 4t + 2v + 3k = d,

(5.41)

Di 6= Dj if i 6= j.

(5.42)

i=1

Ci odd,


62

Remembering the Bianchi Identity, DT a = Ωab ∧ E b , we see that all of these terms vanish if the torsion is constrained to be zero. We can generalise as before: E → E(t), Ω → dt ω(t) + Ω(t), T → dt E(t) + D(t)E(t),

ζ(E, ω) → ζ(E(t), ω(t)).

Now we hit a problem. If the torsion is not constrained to be zero, then D(t)E(t) will not vanish. Neither will D(t)T (t) be expected to vanish. The only obvious way to make d(F ) ζ vanish is to start with a locally exact form in M. Alternatively, we can impose a more complicated constraint on the torsion. Some-times the only constraint needed is to impose the equation of motion involving the torsion. For example, a d-form made from the tensors η, ǫ, Ω, E. Let ζ = Ωa1 ...a2n ∧ Ξ(E)a1 ...a2n

(5.43)

where Ξ is a (d − 2n)-form and n > 0.

Proposition 5.7. The form ζ(E(t), ω(t)) is closed when the field equation from the variation of ω(t) is satisfied (on shell). Proof: The Euler variation w.r.t. ω(t) gives the field equation: n X i=2

= 0. [dt ω(t) + Ω(t)]a3 ...a2k ∧ DΞ(E(t))a3 ...\ a2i−1 ac 2i ab...a2k

(5.44)

By the Bianchi identity, the exterior derivative of L vanishes on shell:

dζ = [dt ω(t) + Ω(t)]a1 ...a2k DΞ(E(t))a1 ...a2k = 0 (on shell).

(5.45)

More generally, when T appears explicitly in Ξ, this on-shell condition is not sufficient. Things are a bit more complicated. This is the subject of further investigations. It is important to notice that (5.45) applies to the Lovelock Lagrangian. If we allow torsion in the Lovelock theory, the thin shell description still applies on shell. If we are interested in a theory of metric and derivatives of the metric (i.e. the same field content as GR), we will set the torsion to zero. This approach seems to lead just to topological terms and the dimensionally continued Euler densities. At the moment, it seems that the Lovelock lagrangian is special in this sense. This would be in accordance with the intuition which comes from Lovelock’s theorem: that the Lovelock Lagrangian is the only one which gives field equations quasi-linear in second derivatives of the metric. This quasi-linearity is obviously important in defining thin hypersurfaces, as discussed in chapter 3. Also the closed form approach is a way of defining thin hypersurfaces. This consideration leads to the interesting conjecture:


63

Conjecture 5.8. If the torsion is constrained to vanish and η(E(t), ω(t)) is a closed form in W , such that η(E, ω) = L(E, ω) on M then L must be either a locally exact form or a sum of dimensionally continued Euler densities. The quasi-linearity of the quadratic Lovelock theory is also important in establishing that it is ghost free about a flat background. We might also conjecture: d(F ) η = 0 ⇒ L gives a theory which is ghost-free about a flat background (?) The converse is definitely not true. A generic Lagrangian cubic in the curvature will be free of ghosts even if there is no corresponding closed form.

6. AN EXPLICIT EXAMPLE As an example, we will look at a co-dimension 2 intersection of N hypersurfaces. This may represent an interesting physical application, related to brane physics. We shall further assume that the hypersurfaces and intersection are not null. We then find explicitly the Junction condition for the intersection with the n = 2 dimensionally continued Euler density (Gauss-Bonnet) term. We also express the energy conservation in the form of relations among the energy tensors involved in this case. Intersections and collisions have been extensively studied for the case of ’s theory [28, 29, 51, 7]. In particular, Langlois et al [51] treated colliding shells in AdS black hole backgrounds. They reduced the problem to adding rapidities just as in special relativity. However, there were some outstanding questions. An important assumption was that upon traversing an infinitesimal loop round the intersection, there is no overall Lorentz boost. This is basically the same as saying there is no deficit angle. Why no possibility of a deficit angle at the intersection? Intuitively, this would be associated only with matter localised on the intersection. We shall see that for Einstein’s theory this is true. For the general Lovelock theory things will be different. It is this difference which is the main physical result of this work.

6.1 The N hypersurface intersection We have a non-simplicial intersection, of co-dimension 2. One way to deal with this is to use the method mentioned in chapter 5. Another way is to have a central cylindrical region and then shrink this region to zero1 . We divide the space-time into N + 1 regions formed by N surfaces intersecting a cylinder in the middle. Taking the cross section of the system, we see a circle with N outgoing lines, without further intersections (fig. 6.1). Call the connection inside the circle ωI , associated with the metric γ. There are N co-dimension 2 simplicial intersections. We calculate the contributions at the intersections, and then take the limit whereby they all coincide i.e. we are going to take the limit of the circle to zero size. γ is chosen to be the induced metric of the resulting co-dimension 2 intersection, I. 1

These methods are equivalent. We can see this by the following: Up to total derivatives, we can eliminate ωI from (6.1). Adding trivially a set of terms L(ωi ω1 ω) + L(ω1 ωi ω) = 0, i = 3..N and using the composition rule we have L(ω1 ω2 ω3 ) + L(ω1 ω3 ω4 ) + .. + L(ω1 ωN −1 ωN ) plus an exact form containing ωI . For N=4, this agrees with (5.22).

6. An explicit example

65

0 2

N

1

Fig. 6.1: The non-simplical intersection viewed as the limit {0} → I, where I is a codimension 2 surface.

The action terms due to the walls are: Z Z Z L(ω1 , ω2 , e) + L(ω2, ω3 , e) + .. + 12

23

N1

L(ωN , ω1 , e)

and, from the k = 3 terms in (4.25), we have the intersection contributions to the action: Z L(ω1 , ω2 , ωI , e) + L(ω2 , ω3 , ωI , e) + · · · + L(ωN , ω1 , ωI , e). (6.1) I

In order to calculate the equation of motion explicitly in terms of intrinsic and extrinsic curvature tensors we should introduce the connection ωij at the common boundaries. This is the connection associated with the induced metric h on the wall {ij}. We are interested in the dimensionally continued theory. By the argument in Section 5.3, the composition rule has validity on the hypersurfaces and intersection. Using the composition rule: L(ωi, ωj , e) = L(ωi , ωij , e) + L(ωij , ωj , e) − dL(ωi , ωj , ωij , e)

(6.2)

we obtain some more contributions at the intersection. Combining the term from the total derivative in (6.2) with (6.1) we get terms: L(ωi, ωj , ωI , e) − L(ωi , ωj , ωij , e).

(6.3)

Using the composition rule again: L(ωi, ωj , ωI , e) − L(ωi , ωj , ωij , e) = L(ωi, ωij , ωI , e) − L(ωj , ωij , ωI , e) (plus an exact form which we can neglect). Substituting this into (6.3)we obtain finally Ld−2 = (12) + (23) + ... + (N1) ;

(ij) ≡ L(ωi , ωij , ωI , e) − L(ωj , ωij , ωI , e). (6.4)


66

Now we can express everything in terms of the bulk region connections and two types of second fundamental form: θi|ij of the surface {ij} induced by the region i; and the θ˜ij of the intersection induced by {ij}.2 θi|ij = ωi − ωij ,

θ˜ij = ωij − ωI .

Their definition in terms of the associated normal vectors and extrinsic curvature tensors are given in the appendix. In order to write the simplest non trivial equation of motion for the common intersection of N (d − 1)-dimensional surfaces we consider the n = 2 dimensionally continued Euler density. Applying (4.5) or (4.18) we find easily L(ωi ωij ω, e) = f (θi|ij , θ˜ij , e).

(6.5)

We assume a gravity Lagrangian of the form LG = β0 L(0) + β1 L(1) + β2 L(2)

(6.6)

where L(n) is the (dimensionally continued) n-th Euler density and β2 is constant of dimensions (length)2 , the coupling of the Gauss-Bonnet term. This theory is sometimes called the Einstein-Gauss-bonnet theory. Varying the frame E a we obtain the equations of motion. We express the second fundamental form θab in terms of the usual extrinsic curvature tensor Kab by equation (B.13) θab = 2n·n n[a Kcb] E c where nµij is the normal vector of a (d − 1)-dimensional surface,{ij}, with orientation induced by the bulk region i. The i, j indices have been suppressed for the sake of sanity. θãb is defined similarly for v µ , the normal vector of the intersection embedded in a given (d − 1)-dimensional hypersurface. We define the corresponding extrinsic curvature tensor: Cµν = γµρ hσν ∇ρ vσ with γµν = hµν − ǫ(v)vµ vν . We have θãb = 2v · v v [a Ccb] E c . Plugging this into (6.5) and (B.12), we get the junction condition for the intersection: X 1 ab ab ab ab ¯ ¯ ¯ ¯ 2∆ (KC) + (KC) − γ Tr(KC + KC) = T˜(d−2) (6.7) 2

where Kab = γca γdb K cd is the projection of K ab onto the tangent space of I. The sum in (6.7) is over all terms in (6.4), one for each hypersurface. We use the notation ¯ ab = Kab − γab K, where K = γ ab Kab , and compact matrix multiplication, for exK ¯ ab = K ¯ ca C cb . ∆{· · · }ij ≡ {· · · }ij − {· · · }ji where {ij} corresponds to the ample (KC)

Note that θ˜ij is defined as the Second fundamental form of I with respect to the wall {ij} not the bulk. 2


67

ab positive orientation. Td−2 is the energy momentum tensor which is localised at the intersection. Notice that it was not necessary to introduce the intrinsic curvature of the intersection C. It is always possible to write everything in terms of the intrinsic and extrinsic curvatures of the walls only. This is discussed briefly in the Outlook section.

6.2 Energy conservation at the intersection Let us see the implications of these results for the question of energy conservation. We recall that the local expression of the energy-momentum tensor conservation is related to the diffeomorphism invariance of the action, under which the metric tensor changes as δgab = 2∇(a ξb) where ξ a = δxa (x) are infinitesimal coordinate transformations. Note that 2∇(a ξb) = δgab has to be continuous. In the previous chapters we have seen that the variation of the hypersurface action gives terms on the intersection through application of Stokes theorem. For non-null hypersurfaces, we can use Gauss’ law. Let us first consider the case of an intersection whose action term is zero. Let N regions intersect, labelled by i, at a common intersection I. We write the energy exchange relations in the system as Z XZ 1 X ab ab δξ Smatter = e Td ∇a ξb + e˜ Td−1 ∇a ξ b = 0 (6.8) 2 ij i i,j=i±1 i where the normal vectors obey nij = −nji , j = i ± 1. Then by ξb = ξ||b + (n·n)nb nc ξc with ξ||b = hcb ξc we obtain XZ ab − e ∇a T(d) ξb + (6.9) i

+

XZ ij

+

ij

+

I

e˜

ij

XZ Z

i

e˜

ij

e˜(d−2)

ab c (n·n)na T(d) hb

ab na T(d) nb

X1 ij

2

1 ab ξc − Da T(d−1) ξb + 2

1 ab c n ξc + T(d−1) K(n)ab (n·n)n ξc + 2 c

ab (v·v)va T(d−1) ξb = 0

where na , Kab are assumed to carry an index ij. Also j = i ± 1; the same for v a which is the normal on I induced by ij pointing outwards. Along with the known relations [22], we obtain then the ones related with the intersection X X ab ab ǫ(v) va Td−1 γbc = 0 , va Td−1 vb v c = 0 (6.10) where the sum is over all shared boundaries. γab is the induced metric at I. Equation (6.10) implies that the total energy current density at the intersection or collision is zero. This is valid though when the energy tensor at the intersection


68

vanishes identically. On the other hand, as we have learned, the energy tensor is not zero in general and the energy conservation has to take into account this lower dimensional energy tensor existing at the intersection hypersurface. In such a case there is an additional term in (6.8) that can be written as Z X 1 ab e˜(d−2) Td−2 ∇a ξ b 2N I ij where we sum over the contribution from each side of every shared boundary for ab N regions. Td−2 is the total energy momentum tensor on I. We decompose ξb = c ξ||b + (n·n)nb n ξc + (v · v)vb v c ξc where ξ||b = γbc ξc . We then have # " Z X 1 ab ab ab {(n·n)Kab nc + (v·v)Cab v c } ξc e˜(d−2) Da (Td−2 ξ||b) − Da Td−2 ξ||b + Td−2 N ij I

(6.11)

(the first term is only useful when the intersection is not smooth itself). The energy exchange relation are then X ab ac (v · v) va Td−1 γbc = Da Td−2 , (6.12) X X ab ab 1 {(n·n)Kab nc + (v·v)Cab v c } = 0 vb v c + Td−2 va Td−1 N ij where the first sums are over all hypersurfaces.

6.3 Colliding Branes and deficit angles For a collision of hypersurfaces, the intersection surface will be space-like. The v vectors are time-like (velocity) vectors. We assume that the hypersurface matter is a fluid described by: v a v b Tab = ρ γca Tab v b = 0. The first of (6.10) is satisfied automatically whilst the second becomes: X ρΛ vΛa = 0

(6.13)

(6.14)

Λ

where the upper case Greek index counts the hypersurfaces. We can recover the results of Langlois, Maeda and Wands [51] by first introducing the ortho-normal basis at the intersection. The basis is taken to line up with the two vectors vI and nI of one of the hypersurfaces. E(0) = vΛ ,

E(1) = nΛ .

We can write the other v vectors in the following way, motivated by Special Relativity, vΞ = γΞ|Λ E(0) + γΞ|Λ βΞ|Λ E(1)


69

where the β and γ have the usual interpretation from SR: βΞ|Λ is the relative speed between the two hyper-surfaces; from the normalisation of v we see γΞ|Λ = (1 − 2 βΞ|Λ )−1/2 . The two components of equation (6.14) are:

X

X

ρΞ γΞ|Λ = 0,

(6.15)

ρΞ γΞ|Λ βΞ|Λ = 0.

(6.16)

Ξ

Ξ

These are the results found in [51]. They are the conservation of energy and momentum respectively. The hypersurfaces obey the same rules in terms of the local inertial frame as do point particle collisions in two dimensions. This is true for quite general bulk backgrounds. The only essential feature is the absence of a deficit angle at the collision. This means that there is a well defined local inertial frame at the collision and the SR addition of velocities applies. We have calculated the contribution to the energy-momentum tensor at the collision due to the junction conditions. Our calculation implicitly assumed that there was no conical singularity (see definition 4.4). There may be some correction to this from a conical singularity. If we impose some reasonable energy condition such as the weak energy condition, this space-like matter should vanish- the two contributions should cancel. The assumption of no conical deficit is then justified for the Einstein theory, because we have seen that there is no contribution due to the junction conditions. But this would not be so for the Gauss-Bonnet theory. In that case, the cancellation would demand that there be a conical singularity at the collision. Conversely, if we impose that there be no such singularity, we must either have space-like matter localised at the collision or stringent selection rules on how hypersurfaces can interact through collisions.

6.4 A three-way intersection in AdS We have seen that there is a possibility to localise matter on an intersection in the Gauss-Bonnet theory. But it remains to be seen whether there is an actual solution to the bulk field equations which gives non-zero energy-momentum. This chapter is devoted to finding examples. 6.4.1 The bulk vacuum solution We shall assume the simplest kind of bulk solution- a constant curvature space-time. 1 A constant curvature space-time satisfies Rµναβ = 12 R(gµα gνβ − gµβ gνα ), R being a constant [41]. This is also known as a maximally symmetric space-time. There are three possibilities: i) de Sitter space (R > 0), ii) anti-de Sitter space (R < 0), iii) flat space (R = 0).


70

In the Einstein theory, empty space will be one of the above three, depending on whether the cosmological constant is positive, negative or zero. In the EinsteinGauss-Bonnet theory it is possible that more than one type of constant curvature space-time will satisfy the vacuum field equations. The different possibilities are because the Gauss-Bonnet is quadratic in the curvature. Anti-de-Sitter (AdS) space has constant negative curvature. 1 Ωab = − 2 E a ∧ E b . l

(6.17)

The constant l has dimensions of length. As already mentioned, AdS space is a vacuum solution of the general Lovelock theory [d/2]

X n=0

n bn

an Ωa1 b1 ∧ · · ·Ωa

∧ ea1 ···bnc = 0

provided that the following relation is satisfied:3 [d/2]

X (−1)n d! an = 0. 2n (d − 2n)! l n=0

We will write the AdS metric in conformally flat form: ds2 =

1 ηµν dxµ dxν , ((u · x)/l + 1)2

(u · x)/l + 1 ≥ 0,

(6.18)

where u · x ≡ ηµν uµ xν and u is a constant normalised to ηµν uµ uν = 1. We will only be interested in the vicinity of the intersection and will not worry here about the global details of joining together regions of AdS. We will work in the moving frame formalism and choose the ortho-normal frame field: E (µ) =

1 µ dx , J

J (x) ≡ (u · x)/l + 1.

We use the zero Torsion condition: dE (µ) = −ω

(µ) (ν)

∧ E (ν) .

Evaluating the left hand side: 1 ηνρ uρ ν 1 ηνρ uρ (µ) ∂ ν µ µ dx ∧ dx = − dx ∧ dx = E ∧ E (ν) . ∂xν J J2 l l 3

(6.19)

(6.20)

An interesting choice of coefficients is that of Chamseddine’s Gauge theory of the AdS group. The Gauge symmetry fixes all but two of the coefficients l and κ. The three terms in the five dimensional theory are: (d − 1)l2 (d − 1)(d − 3)l4 κ 1 f (E ∧d ) + f (Ω ∧ E ∧(d−2) ) + f (Ω∧2 ∧ E ∧(d−4) ) . L= d l d 2(d − 2) 8(d − 4) In this case 6.17 is the only constant curvature solution of the vacuum field equations.


71

2

2

3

1

1

Fig. 6.2: The intersection without conical singularity.

Comparing (6.20) and (6.19) and making use of ω ab = −ω ba for a Lorentzian connection, we compute the connection coefficients: 1 (6.21) ω (µ)(ν) = (uµ E (ν) − uν E (µ) ). l 6.4.2 Three-way intersection We will consider the simplest 3-point vertex. There are 3 regions Region 1: 0 < θ < θ1 , Region 2: θ1 < θ < θ2 , Region 3: θ2 < θ < θ3 , with the identification θ3 ≡ 0. One can have a conical singularity at the intersection with deficit angle 2π − θ3 but we will not do so for reasons already mentioned. so we take θ3 = 2π (fig. 6.2). Let us work using the co-ordinates y, z for the plane. x = ρ cos θ, y = ρ sin θ. In each region i let ui = (0, ..., 0, cos φi , sin φi) such that ui · x = ρ cos (θ − φi ). The metric in each region takes the following form: 1 ds2i = ηµν dxµ dxν . (6.22) (ρ cos(θ − φi)/l + 1)2 The metric should be continuous across the walls:

cos(θ1 − φ1 ) = cos(θ1 − φ2 ), cos(θ2 − φ2 ) = cos(θ2 − φ3 ), cos(φ3 ) = cos(φ1 ). cos a = cos b of course has two solutions, a = ±b. If we are to have any matter on the walls we must choose the non-smooth minus sign solutions: φ1 = −φ3 = θ1 − θ2 , φ2 = θ1 + θ2 .


72

The contribution to the hypersurface from the Einstein gravity is −β1 δE c ∧ (ω2 − ω1 )ab ∧ eabc = 2(d − 2)

α1 (u2 − u1 )a δE c ∧ eac . l

(6.23)

Above, we have used E b ∧ eabc = −(d − 2)eac . Now, since (u2 − u1 ) · x = 0 on the hypersurface, (u2 − u1 ) must be normal to the hypersurface. The norm of this vector with respect to ηab is: p |(u2 − u1 )| = 2 − 2 cos(φ2 − φ1 ) so

p (u2 − u1 )a = ± 2 − 2 cos(2θ2 ) na .

p Similarly (u3 − u2 )a = ±na{23} 2 − 2 cos(2θ1 ) and also a similar relation for the p hypersurface {31}: (u1 − u3 )a = ±na{31} 2 − 2 cos(2θ1 − 2θ2 ). Substituting this into (6.23), the hypersurface term due to the Einstein theory is: ±2(d − 2)

α1 a p n 2 − 2 cos(2θ2 ) δcb δE c ∧ eab . l

By (B.12), the energy momentum on the brane in the Einstein theory is: (T{12} )ab = ∓(d − 2)

α1 p 2 − 2 cos(2θ2 ) δba . l

(6.24)

So there is indeed hypersurface matter. It takes the form of a (d − 1)-dimensional cosmological constant. With the Gauss-Bonnet term this will get modified. We will not calculate this but will proceed to find the intersection term. From equation (4.32), this term is: Z d2 t(ω2 − ω1 )ab ∧ (ω3 − ω2 )cd ∧ δE f ∧ eabcdf . β2 A2 s123

Substituting A2 = −2 and 1/2 for the volume of the simplex: − β2 δE f ∧ (ω2 − ω1 )ab ∧ (ω3 − ω2 )cd ∧ ef abcd

= −4β2 (u2 − u1 )a (u3 − u2 )c δE f ∧ E b ∧ E d ∧ ef abcd = 4β2 (u2 − u1 )a (u3 − u2 )b δE f ∧ E c ∧ E d ∧ ef abcd

= 4(d − 4)(d − 3)β2 [(u2 − u1 )(1) (u3 − u2 )(2) − (u2 − u1 )(2) (u3 − u2 )(1) ]δE f ∧ef (1)(2) .

The factor in square brackets is (cos φ2 − cos φ1 )(sin φ3 − sin φ1 ) − (cos φ3 − cos φ1 )(sin φ2 − sin φ1 ) = sin(φ2 − φ3 ) + sin(φ3 − φ1 ) + sin(φ1 − φ2 ) = sin(2θ1 ) + sin(2θ2 − 2θ1 ) − sin(2θ2 ).


73

since e(1)(2) is the natural volume element on the intersection, we have, for the matter localised there: (T˜123 )ab = 2(d − 4)(d − 3)β2 [sin(2θ2 ) + sin(2θ1 − 2θ2 ) − sin(2θ1 )]δba .

(6.25)

This is a (d − 2) dimensional cosmological constant. This term will vanish if: i) θ1 = θ2 , θ1 = 0 or θ2 = 0 (two of the walls coincide); ii) θ1 = π, θ2 = π or θ1 − θ2 = π (there is a smooth wall with another branching off). None of these cases are a genuine 3-way intersection: either two hypersurfaces coincide or the matching of the metric is smooth across a hypersurface. Thus we conclude that for this example, localised matter at the intersection is inevitable. There are many other ways to have three walls intersecting in an AdS bulk. The above is the simplest case of static walls with cosmological constant type matter. More general solutions have been found by Lee and Tasinato [52].

7. CONCLUDING REMARKS AND OUTLOOK We have found that Lovelock gravity of any order, nmax , in any space-time dimension, d, has a well defined description of thin hypersurfaces of matter. This is obtained from an action principle which yields junction conditions for each hypersurface and intersection. The junction conditions relate the discontinuity in extrinsic curvatures as well as the intrinsic curvatures to the localised energy-momentum. For intersections of co-dimension greater than nmax these junction conditions are trivial- there is no localised matter. We have proved our results by two different ways of interpolating. As well as providing a useful check that the numerical factors are correct, the two approaches have different advantages. The method of chapter 4 was used to establish that the action with surface terms is one and a half order. The method of chapter 5 was used to prove the formal equivalence of this action with the Lovelock action. The results have been found in terms of the vielbein and connection 1-forms. It would be useful to express them in terms if the extrinsic and intrinsic curvature tensors. We can give a simple argument: e is a (d − n)-dimensional volume element. The Lovelock term of order n is: Rn ∧ e;

(7.1)

˜ + K 2 )n−1 ∧ e; K ∧ (R

(7.2)

˜ + K 2 )n−2 ∧ e; K 2 ∧ (R

(7.3)

Kn ∧ e

(7.4)

At a co-dimension one brane:

and co-dimension 2:

co-dimension n:

when we run out of R’s. This is very schematic: K represents a sum of extrinsic ˜ denotes a sum over intrinsic curvatures. curvature terms with some coefficients; R More accurately, using the formulae of B.3, the Action terms are: Z Z a ...a2p a2p+1 ...a2n p dt S∝ e˜ (∆K)[ap+1 R(t)a2p+1 (7.5) ...a2n ] p+1 ...a2p s0..p

{0...p}

where e˜ is the volume element on the hypersurface, Ω(t)ab ≡ 1/2R(t)abcd E c ∧ E d and p+1 2p (∆K)ap+1 ...a2p ≡ (K{1|10} − K{0|10} )cap+1 · · · (K{p|p0} − K{0|p0} )ac2p .

7. Concluding remarks and outlook

The Junction tensors are: Z ap+1 ...a2p a2p+1 ...a2n bbp+1 ...b2n a dp t δaa (∆K)bp+1 Gb ∝ ...b2p R(t)b2p+1 ...b2n . p+1 ...a2n

75

(7.6)

s0..p

Finding the exact numerical factors and performing the general integral over the ˜ + K 2 (see simplex are work in progress. R(t) can be decomposed into terms ∼ R equation 5.32). As mentioned at the end of chapter 3, the action may not, it seems, represent an unambiguous limit of a smooth matter distribution. It is none-the-less a selfconsistent theory of strictly zero thickness hypersurfaces. In chapter 5 we have found a tidy way of expressing all the terms as Z ηDC , W

the intersection terms coming from the expansion of a simple polynomial. This polynomial is given in equation (5.28). The meaning of this is not yet fully clear.

7.1 Intersecting or colliding braneworlds Intersections can be different types according to the signature of the induced metric: i) ii) iii)

positive definite, negative definite, indefinite,

space-like (collision). time-like intersection. can change between time-like, spacelike or null.

I will only discuss the first two cases here since the physical meaning is more clear. An intersection of hypersurfaces of a given geometry which in GR has no localised matter, will generally have localised matter coming from the higher order Lovelock terms. i) For a collision we can demand no localised space-like matter and interpret the constraint on the geometry as conservation of energy. Even if the coefficients of the higher order terms are small, this constitutes a qualitative difference between GR and the higher order Lovelock theories. ii) An important model in String theory is (chiral) matter localised on time-like intersections of branes. The low energy effective theory would be expected to include higher curvature terms. With the higher curvature terms in the Lovelock form, we can have localised matter classically, not as a black hole but as a kind of defect, matching several vacuum regions of space-time. It should be pointed out that in, say, d = 11 dimensions, only the first five Lovelock terms are non-zero, nmax = 5. This allows only intersection brane-worlds of dimension 6 or greater to be realised in this way. To get 4-dimensions, we need more singular branes, such as the intersecting co-dimension 2 branes considered by Navarro and Santiago [67].


76

7.2 Acceptable singularities In Lemma 4.5 it was assumed that there was no localised curvature on surfaces of co-dimension > 1. Indeed, our action principle will not work without some modification if there is such a singularity. In four dimensional GR, point particles with Schwarzschild geometry and thin cosmic strings which produce a conical geometry are examples of these kind of singularities. This section is devoted to a discussion of why the co-dimension 1 singularities are special. 7.2.1 A question of derivatives A crucial feature is the treatment of the derivative of some field σ which is actually discontinuous. The derivative itself is undefined but can it be well defined under R integration? In section 3.4 we outlined a way of meaningfully defining dσ across a wall. At the intersection itself however, the problem seems to be hopeless (see fig. 7.1a). But nonetheless we proceeded to find expressions for the intersection terms. How has this happened? There is a sleight of hand involved. From the point of view of the bulk regions, we treat it as if the intersection has been removed. However, the co-dimension 2 intersection reappears, as ifR by magic, as part of the boundary of each of the walls (fig. 7.1c). In this way, dσ has been given meaning. There are two ways of interpreting this: 1) We have cheated; 2) There is some underlying mathematical description that is rigorous. Strong evidence in support of the latter is found in chapter 5. We saw that the expression for the intersection terms could be written in terms of a singular mapping from some non-singular theory. Future work would be to make precise what is meant by “well defined under integration”. This would involve the mathematics of distributions [18] or generalised functions [50]. In throwing out the derivative shown in figure 7.1, we see the importance of disallowing deficit angles. Then it seems such a derivative can not be ignored. 7.2.2 Conical singularities and the like The cone is a topological manifold. One can overlay the tip of the cone with a co-ordinate region which looks like Rd but in these co-ordinates, the metric will be infinite at the tip. As such there is no sensible way to define an orthonormal frame. The inner product if two vectors is ill defined. This is obvious if one represents the cone by a flat space with a wedge taken out (fig. 7.2). Everywhere except at the tip, one can draw two arrows which are perpendicular. At the tip, whether they are perpendicular or not depends on whether you measure the angle clockwise or anti-clockwise. 1 It is this breakdown of ortho-normality which we shall regard as an 1

There is another possibly well defined situation. If there is a deficit angle of exactly π (fig. 7.3), one can also define unambiguously two orthogonal normal vectors. Since going round the


77

(b)

(a)

111111111111 000000000000 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 000000000000 11111111111111111 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000 000000000000111 111111111111 00000000000000000 00011111111111111111 111

(c)

Fig. 7.1: (a) The derivative across the intersection is undefined. (b) The derivative across the wall can be well defined under integration. (c) The intersection appears as the shared boundary of the walls.


78 O

B

identify OA with OB V’(q)

q

O

p=q

V(p)

V’ V

p A

A=B

Fig. 7.2: The cone. Parallel transport of a vector around the singularity gives a different vector. This means that the relative angle of two vectors at the singularity is ill defined.

unacceptable singularity. It is a breakdown of the d-dimensional equivalence principle since, at the singularity, the tangent space does not look locally like Minkowski space. As well as the cone, or cosmic string, such things as the Schwarzschild singularity will be considered unacceptable for the same reason. Assuming we disallow deficit angles, the localised matter does not come from localised curvature at the intersections. The “delta functions” come from the higher curvature terms (although we don’t really use delta functions- we deal only with Stokes Theorem). Schematically, for an intersection in the x − y plane: f (Ω ∧ E ∧(n−1) ) ≈ Aδ(x) + Bδ(y), f (Ω∧2 ∧ E ∧(n−2) ) ≈ Cδ(x, y).

If there is a conical singularity, there will be curvature at the intersection that is not induced by the hypersurfaces: f (Ω ∧ E ∧(n−1) ) ∼ δ(x, y) and so our method breaks down. 7.2.3 Singular gravity sources The theory of General Relativity admits singular sources of gravity which are hypersufaces. They are particles which, although singular, produce only a mild form of geometrical singularity. Geodesics across the hypersurfaces are well defined. There is a well defined local Lorentz frame at the hypersurface. Hypersurfaces then may be thought of as fundamental particles compatible with classical General Relativity. intersection sends n → −n, n1 · n2 = 0 is well defined.


79

B O V’(q) q Identify OA with OB p=q O V’=−V V(p) p

A=B A

Fig. 7.3: The cone with deficit angle = π.

Generally, where the curvature becomes singular, we expect to be in the territory of quantum gravity. A completely different notion of geometry may be necessary there or new types of field theories. So it may be that this preferred status of hypersurfaces, being classical, is irrelevant. However, it is conceivable that “quantum gravity” is actually a classical theory [81] 2 . Even if it is not, the distinction between hypersurfaces and other singular particles may still prove to be of value. There is also a sense in which co-dimension 1 and co-dimension 2 membranes are special. In both cases, one can have delta function singularities in the Einstein tensor without the Weyl tensor diverging near the membrane. The membrane is not surrounded by a black hole but is a kind of defect in space-time. In the Einstein-Gauss-Bonnet theory, it was shown by Bostock et al [10] that it is possible to have a co-dimension 2 source (Braneworld) where the gravity singularity comes only from the Gauss-Bonnet term. In this case there is not a deficit angle. The problem with this is that the Riemann tensor diverges in the vicinity of the brane. This led them to disregard it since the quantum effects would be expected to be large in the region of the brane. 7.2.4 Gravity on simplicial manifolds There are several approaches to quantum gravity which involve discretising the geometry of space-time at some small (Planck) scale. Space-time is divided into cells, with the curvature concentrated on the edges of the cells. As well as being a simple model, this is reasonable because it is expected that the structure of space-time itself be very different on the Planck scale. The effects of quantum fluctuations on the 2

For an interesting article on the theological implications of non-determinism see [12].


80

curvature of a smooth space-time are so problematic that it has been suggested that the smooth manifold structure breaks down at small length scales. One would like to have an action principle for Lovelock gravity on any triangulated manifold involving boundary terms. The trick is to take account of the manyvaluedness of the connection and the normal vectors at the singularities, so as to get an unambiguous result. For example, let us assume that an intersection coincides with the tip of the cone in fig. 7.2. Let Eia be the vielbein in one of the bulk regions adjacent to the singularity. Going once round the singularity will meanPEia → Λab Eib , where Λ is some Lorentz transformation. Simply defining E a (t) = i ti Eia is not sufficient. There is a similar problem with the connection. The fact that one has metric and connection (or vielbein and connection 1-form) degrees of freedom seems to be a source of complication. Chamseddine’s formulation of certain Lovelock theories as gauge theories [14] (involving only connection degree of freedom) may help in this regard. This formulation is valid in any odd number of dimensions. In four dimensions, there are the Ashtekkar variables to describe GR in terms of purely affine connection degrees of freedom, which are important in Loop quantum gravity [72]. A general purely affine formulation of Lovelock gravity in even dimension is not known at the present.

7.3 Geometrical conundrums 7.3.1 Non-simplicial intersections The treatment of a simplicial intersection was found to be especially easy. That being said, there is no great problems with treating non-simplicial intersections- it is just a matter of summing terms in the correct way, according to the principles of Section 5.2.3. Let us denote a co-dimension p (non-simplicial) intersection by I p . Conjecture 7.1. The intersection lagrangian of I p is determined solely by the set of I 1 ’s for which I 1 ∩ I p 6= ∅.

That is to say, in order to know what sum of terms appears in the intersection, it is sufficient to know which hypersurfaces meet at that intersection. If this conjecture is true, it would be easy to write down the action for any kind of intersection- the way of summing terms is determined just by the list of hypersurfaces without needing any information about intermediate co-dimension intersections. 7.3.2 Mappings In section 5.3 we have found the action in the form of an integral over the manifold we called W . W lives in a higher dimensional manifold F which looks locally like the product M × S N , S N being a simplex. The action is non-singular in F . It is the projection W → M which is singular. We could view E(t) and ω(t) as fields on F . They are independent fields but, by definition, the fields, pulled back to W obey: d(F ) E(t)a + ω(t)ab E(t)b = 0,

(7.7)


81

by equations (5.26) and (5.27). This is some kind of zero torsion condition. What does it mean? It would be interesting to find out. 7.3.3 The dual lattice The simplicial complex which results from taking the quotient: W M tells us which regions are connected to which. Each bulk region is represented by just a point. A hypersurface is represented by a line between the two bulk points. Intersections are represented by higher dimensional shapes with the bulk points as vertices. The result is a dual lattice to the original. For a simplicially valent lattice, these shapes will be simplices. As such, W/M is a kind of triangulation of M. This is a strange fact- by suppressing all the x-directions we end up with something of exactly the same topology as M.

APPENDIX

A. SOME MATHEMATICAL PRELIMINARIES This chapter contains useful background material, presented in a notation consistent with the main chapters. More details can be found in the relevant textbooks.

A.1 Extrinsic curvature This section is based on Wald [85], but we use the opposite sign convention for the extrinsic curvature. We have a hypersurface Σ. In a surrounding region O we can have a congruence of geodesic curves. This is a family of geodesics which pass through each point in O once and only once. The tangents form a vector field in O. We will take ζ to coincide with a normal vector on Σ. Let ζ be the vector field ζa =

dxa , dλ

where λ is the proper time or proper distance along a geodesic for time-like or spacelike curves respectively. The vectors are normalised ζ a ζa = ∓1

(A.1)

with the −/+ sign for time-like or space-like vectors respectively. We define the extrinsic curvature Kab = −∇a ζb .

(A.2)

It follows from the equation for a geodesic and from the normalisation (A.1) that Kab is orthogonal to ζ: ζ a ∇a ζb = 0; 1 ζ b ∇a ζb = ∇a (ζ b ζb) = 0. 2

(A.3)

We define the induced metric hab = gab − ζa ζb ζ ·ζ and also the projection operator ⊥. This is a projection from the tangent bundle T (M) onto the subspace on T (M)|O 1 which is perpendicular to ζ: ⊥ab ≡ hab = g ac hcb ; 1

This subspace is only well defined in the neighbourhood O.

A. Some mathematical preliminaries

ζa (⊥ab v b ) = 0,

84

∀v ∈ T (M).

From the theorem of Frobenius, since ζ is hypersurface orthogonal, we have ζ[a ∇b ζc] = 0. Contracting this with ζ a and using (A.3) we see that the extrinsic curvature is a symmetric tensor: ζ a ζa K[bc] = ∓K[bc] = 0.

(A.4)

In the region O one can always write the metric in terms of Gaussian Normal co-ordinates. One of the co-ordinates, say w is identified with the proper time of the geodesic congruence. In these co-ordinates ζa =

∂xa = (0, ..0, 1) ∂w

and the metric takes the form ds2 = hij dxi dxj ∓ (dw)2 where i, j are from 1...d − 1. hij is the intrinsic metric tensor on Σ. In these coordinates, the intrinsic curvature is 1 Kij = − ∂w hij = (ζ · ζ)Γw ij . 2

(A.5)

We can also define the extrinsic curvature in terms of any unit vector, n, normal to Σ. This vector field will agree with ζ on Σ. And so the tangential derivatives will agree on Σ: hab = gab −

na nb ; n·n

hca ∇c nb = hca ∇c ζb. Using (A.2) and (A.3 we get: hca ∇c nb = −Kab . The above is the most important expression for the intrinsic curvature for our purposes. We can also write the extrinsic curvature as a Lie derivative: 1 Kab = − £nhab . 2


85

A.2 Exterior differential calculus An exterior differential form is basically an anti-symmetrised tensor. Let ψ be an exterior p-form. Then: ψa1 ...ap = ψ[a1 ...ap ] , where the square brackets [ ] denote complete anti-symmetry with respect to the indices. There is an exterior product which preserves this anti-symmetry, the wedge product: (ψ ∧ σ)a1 ...ap+q =

(p + q)! ψ[a1 ...ap σap+1 ...ap+q ] p! q!

where ψ and σ are p and q forms respectively. More abstractly, a p-form is an anti-symmetric multi-linear map from the product of co-tangent spaces Tx∗ M × · · · × Tx∗ M to R. It can be written in terms of the basis: ψ = ψµ1 ...µp dxµ1 ∧ · · · ∧ dxµp . The wedge is the anti-symmetrised product of the basis 1-forms dxµ ∧ dxν = −dxν ∧ dxµ . One can also write ψ in terms of a non-co-ordinate basis of Tx∗ M: ψ = ψa1 ...ap E a1 ∧ · · · ∧ E ap . The exterior derivative, d, takes a p-form into a (p + 1)-form: (dψ)µ1 ...µp+1 = (p + 1)∂[µ1 ψµ2 ...µp+1] . Definition A.1. A p-form, ψ is closed if dψ = 0. A p-form, ψ is exact if there exists a (p − 1)-form, σ, such that ψ = dσ. It will be useful to deal also with tensor-valued differential forms. Let σ be a type (p, q + k) tensor valued exterior differential k-form. If p = q = 0 the exterior derivative of σ is tensorial. If p, q 6= 0 the exterior derivative is not co-ordinate invariant. The appropriate tensorial derivative operator is the exterior co-variant derivative: ...µp ...µp Dσνµ11...ν = dσνµ11...ν + q q

p X i=1

...λ...µp ωλµi ∧ σνµ11...ν − q

q X j=1

...µp ωνκj ∧ σνµ11...κ...ν . q

ω is a connection 1-form. ωνµ = Γµνλ dxλ . It will be helpful to write the exterior co-variant derivative in terms of the tensor notation. ! q+k 1 X λ µ1 ...µp µ1 ...µp µ1 ...µp Dσν1 ...νq = ∇κ σν1 ...νq+k + Tνi κ σν1 ...λ...νq+k dxκ ∧ dxq+1 ∧ · · · ∧ dxq+k . 2 i=q+1


86

µ If the torsion tensor Tνκ is zero, the exterior co-variant derivative is: ...µp ...µp Dσνµ11...ν = ∇κ σνµ11...ν dxκ ∧ dxνq+1 ∧ · · · ∧ dxνq+k . q q+k

In terms of some non-co-ordinate basis E a : a ...a

a ...a

p E c ∧ E bq+1 ∧ · · · ∧ E bq+k . Dσb11...bqp = ∇c σb11...bq+k

ωba = Eµa DEbµ or a ωbc = Eµa Ecν ∇ν Ebµ .

I collect here some useful equations. Let ψ and σ be exterior differential p and q forms respectively. ψ ∧ σ =(−1)pq σ ∧ ψ d(ψ ∧ σ) =dψ ∧ σ + (−1)p ψ ∧ dσ D(ψ ∧ σ) =Dψ ∧ σ + (−1)p ψ ∧ Dσ

(A.6) (A.7) (A.8)

ψ [ab ∧ σ cd] =(−1)pq σ [ab ∧ ψ cd]

(A.9)

ψ

[ab

c]

pq

∧ σ =(−1) σ

[ab

∧ψ

c]

ψ [a ∧ σ b] =(−1)pq+1 σ [a ∧ ψ b]

(A.10) (A.11)

A.2.1 Orthonormal frames Let ξ be a set of basis vectors {ξ1 , ..., ξd } of the vector space Rd . A linear frame u(ξ) = {V1 , ..., Vd } is an ordered set of basis vectors of the tangent space Tp (M). u is a linear isomorphism from Rd to Tp (M). The set of all u(ξ) at each point p in M forms a Principal Fiber Bundle with structure group GL(d, R) [49]. There is a natural Lorentzian metric on Rd , the Minkowski metric η. We restrict ξ to be an orthonormal basis η(ξa , ξb) = ηab , ηab ≡ diag(−1, 1, ..., 1). The orthonormal basis E = u(ξ) = E1 , ..., Ed is again an ordered set of basis vectors of Tp (M) but u−1 also induces a canonical metric on M: g(Ea , Eb ) = ηab .

(A.12)

Conversely, given a metric on M, we can always find a frame E such that A.12 is satisfied. The set of all E at each point p in M forms a Principal Bundle with structure group SO(d − 1, 1), the restriction of GL(d, R) to those transformations which preserve the Minkowski metric. The physical picture is this: The tangent space at any point p on the manifold looks like Minkowski space. E is simply a set of vectors: Ea = Eaµ ∂µ


87

such that gµν Eaµ Ebν = ηab . i.e. E describes a local inertial frame. There is a gauge freedom in the choice of E corresponding to the set of all inertial frames, related by local Lorentz transformations. If space-time is curved, Ea will be a non-co-ordinate basis. i.e. there will be no co-ordinates xa on M such that Ea = ∂a . This expresses the fact that there is no global inertial frame. In this Thesis, I will deal with the co-frames E a which are the dual vectors to Ea . In a corruption of notation, I have referred to them throughout as frames. It should be remembered that they are really co-frames. E a is also referred to as the vielbein. A.2.2 Integration of forms on oriented manifolds The support of a form on a manifold M is the domain in which the form is non-zero: supp ω = {x ∈ M|ω(x) 6= 0}. Definition A.2. Integration on an oriented manifold. Let {U P i , φi} be an open cover of the manifold M. We define a partition of unity ω = that fi is a i fi ω suchP smooth function non-zero only in region Ui and at every point x ∈ M, i fi (x) = 1. We define the integral Z XZ XZ ∗ ω= fi ω = (φ−1 (A.13) i ) (fi ω). M

i

Ui

i

φi (Ui )⊂Rd

The sum is finite if the support of ω is compact. in which case the Ui can be chosen to be a countable basis. Definition A.3. Domain with regular boundary. Let M be an oriented manifold with an atlas {Ui , φi}. A subset D is called a domain with boundary if for each point x ∈ D there is a chart (U, φ) at x such that either: (a) φ(Ui ∩ D) is an open neighbourhood of φ(x) in Rd (Ui is contained within D); or (b) φ(Ui ∩ D) is an open neighbourhood of φ(x) in the half space H = {(x1 , ..., xd ) ∈ Rd | x1 ≤ 0}. In the case (b), the boundary cuts through Ui and is defined by x1 = 0. Definition A.4. Manifold with regular boundary. For the important case D = M, then M is a manifold with boundary. Definition A.5. The orientation on ∂D induced by M. Let the orientation on M be dx1 ∧···∧ dxd > 0. Then the orientation defined on the surface ∂D (xi = 0 ∈ φ(∂D)) is ci · · · ∧ dxi > 0. (−1)i−1 dx1 ∧ · · ·dx

(A.14)

We use an arbitrary co-ordinate xi . This will be useful later in the case of domains with irregular boundaries. We have defined xi ≤ 0 to be the interior of D. If we had defined it the other way round, the minus sign factor in (A.14) would be (−1)i .


The integral over a section of the boundary is Z Z 1 d i−1 c i dx · · · dx · · · dx = (−1) φ(∂D∩U )

(−1)i−1

Z

b1

···

a1

Z

φ(D∩U )

bd

ad

88

ci · · · dxd |xi =0 dx1 · · · dx

ci · · · dxd |xi =0 . dx1 · · · dx

Theorem A.6 (Stokes’ Theorem on an Oriented Manifold). Let D be a domain in M, ω a (d − 1)-form in M such that supp ω ∩ D is compact. Z Z dω = i∗ ω. (A.15) D

∂D

Proof: Using the definition (A.13) of integration on a manifold: Z XZ ∗ iω= i∗ (fi ω); ∂D

Z

i

dω =

D

XZ i

∂D

d(fi ω). D

So in order to prove Stokes’ Theorem, one just needs to prove Z Z ∗ i (fi ω) = d(fi ω), ∀i. ∂D

Let

∗ (φ−1 i ) (fi ω) =

(A.16)

D

X j

cj · · · ∧ dxd . aj (x) dx1 ∧ · · · dx

The support of fi ω is contained within Ui The functions fi are smooth so fi → 0 on the boundary of Ui . If Ui does not contain the boundary, Ui ∩ ∂D = ∅, the right hand side of (A.16) can be seen to vanish as required. Z Z X ∂aj (x) 1 j−1 d(fi ω) = (−1) dx · · · dxd j ∂x φi (Ui ) D j XZ cj · · · dxd [aj (xj = max) − (xj = min)] = 0. = dx1 · · · dx j

If Ui intersects the boundary, Ui ∩ D = {(x1 ...xd ∈ Rd |x1 ≤ 0}, then the integral does not vanish as above. The integrals over x2 , ...xd vanish but the integral over x1 , in general not being over the whole of supp fi ω, may not vanish. Z Z X ∂aj (x) 1 j−1 d(fi ω) = (−1) dx · · · dxd (A.17) j ∂x φ (U ∩D) D i i j Z = a1 dx2 · · · dxd 1 =

Zφi (Ui ∩D)

φi (Ui ∩∂D)

x =0

a1 dx2 · · · dxd .


89

The left hand side of A.16 manifestly vanishes if Ui ∩ ∂D = ⊘. On the other hand, if Ui ∩ ∂D 6= ⊘ Z Z ∗ i (fi ω) = a1 (x1 = 0) dx2 · · · dxd . (A.18) ∂D

φi (Ui ∩∂D)

Comparison of A.17 and A.18 verifies A.16 and completes the proof of Stokes theorem. Definition A.7. Domain with regular p-corners. [78] Let M be an oriented manifold with an atlas {Ui , φi }. A subset D is called a domain with regular p-corners if for each point x ∈ D there is a chart (U, φ) at x such that φ(Ui ∩D) is an open neighbourhood of φ(x) in either Rd or an s-corner: S = {(x1 , ..., xd ) ∈ Rd | x1 , ..., xs ≤ 0} for s=1...p. An s-corner is where s regular sections of boundary meet at x1 = x2 = ... = xs = 0. Some polygons have faces which are not described by the s-corner prescription but can be broken into pieces which do. We now extend the validity of Stokes’ theorem (A.15) to a domain with regular p-corners. For patches which do not intersect the boundary or which intersect regular sections of the boundary the same argument as before applies. We now need to argue (A.16) for the case where φi (Ui ∩ D) is an s-corner, s ≥ 2. Z Z X ∂aj (x) 1 j−1 dx · · · dxd d(fi ω) = (−1) j ∂x φ (U ∩D) D i i j Z s X dk · · · dxd . (−1)k−1 ak (xk = 0) dx1 · · · dx = =

Zk=1

i∗ (fi ω).

∂D

In the last line (A.14) has been used. On an oriented manifold we define the d-dimensional volume element: p 1 e = ǫa1 ...ad E a1 ∧ · · · ∧ E ad = E 1 ∧ · · · ∧ E d = | det(g)|dx1 ∧ · · · ∧ dxd . (A.19) d!

We also define a (d − r)-dimensional volume element: ea1 ...ar =

1 ǫa1 ...ad E ar+1 ∧ · · · ∧ E ad (d − r)!

(A.20)

e.g. e(1)...(r) = ǫ(1)...(d) E (r+1) ∧ · · · ∧ E (d) . This is a volume element on a hypersurface with the choice of adapted frames {E (1) , ..., E (r) ; E (r+1) , ..., E (d) } where {E (1) , ..., E (r) } forms a basis of the space of normal vectors and {E (r+1) , ..., E (d) } forms a basis of tangent vectors. ea1 ...ar will


90

also be useful in writing the dimensionally continued Euler densities. Useful formulae: b

δea1 ...ar = δE b ∧ea1 ...ar b

E ∧ea1 ...ar =

(A.21)

r!e[a1 ...ar−1 δabr ]

(A.22) b

...b

r na1 ...ap E bp+1 ∧· · ·∧E br ∧ea1 ...ar c = (−1)r (r − p + 1)!na1 ...ap ∧ea1 ...ap [c δap+1 (A.23) p+1 ...ar ]

for na1 ...ap a rank p tensor, p, q, r ≥ 0, r > p.

We will define the canonical volume element on a boundary in such a way that Gauss Law takes its usual form. Let σ = σ µ eµ . dσ = (∇µ σ µ )e. Z

σ = (n·n)

∂M

Z

σ µ nµ nν eν .

∂M

So Gauss’ Law takes the usual form: Z Z µ (∇µ σ )e = M

with the convention:

σ µ nµ e˜

(A.24)

∂M

√ e˜ = (n·n)nν eν = (n·n) hdd−1 x.

(A.25)

A.2.3 Integration over simplices and chains Definition A.8. A standard p-simplex, Sp in Rp is defined, following von Westenholz [84], by p ) ( X xi ≤ 1; 0 ≤ xi ≤ 1 ∀i (A.26) sp = (x1 , ..., xp ) i=1

i.e. the standard p-simplex is the closed convex hull (A0 , , ..., Ap ) of (p + 1) points, taken in a definite order and such that the p vectors (Ai − A0 ), i = 1...p, are linearly independent. Examples: standard standard standard standard

0-simplex: 1-simplex: 2-simplex: 3-simplex:

a point, s0 = {0}. a unit interval, s1 = [0, 1]. an oriented triangle with points at (0,0), (1,0), (0,1). an oriented tetrahedron.

The boundary of a standard p-simplex is the oriented sum of the (p − 1)-dimensional faces: p X ck , ..., Ap ). (−1)k (A0 , ..., A (A.27) ∂sp = k=0


91

Fig. A.1: The standard simplices for 0,1,2 and 3 dimensions.

Definition A.9. A differentiable singular p-simplex, σp on a differentiable manifold M is given by a C ∞ map of a standard p-simplex into M, f : sp → M. To be differentiable at the boundaries, it must be possible to extend the map to f : U → M, where U is an open subset of Rp containing sp . Under such a map, σp is the image of sp in M. It is possible to cover Rp with sp , i.e. they provide a triangulation of Rp . The σp do not in general provide a triangulation of M. Definition A.10. A differentiable p-chain, cp in M is defined as the formal sum of a set of p-simplices {σpi } with real coefficients. X cp = λi σpi , λ ∈ R. (A.28) i

The boundary of a p-simplex is ∂σp =

p X

(−1)k σpk

(A.29)

j=0

ck , ..., Ap ) in M. where k denotes the k-th face of the simplex, the image of (A0 , ...A

Theorem A.11 (Stokes’ Theorem for a simplex). Let σd be a differentiable d-simplex with boundary ∂σd in a d-dimensional differentiable manifold M. Let ω be a differentiable (d − 1)-form. Then Z Z dω = ω. (A.30) σd

∂σd

The proof is in the textbooks [84].

A.3 The Euler number The Euler number, or Euler-Poincare Characteristic, χ(M), is a topological invariant of the manifold. It is preserved under a homeomorphism, a global 1-to-1 map from


92

Homeomorphic

Fig. A.2: The coffee cup and the donut have the same Euler number, χ = 0

one manifold to another. For example, the donut and the coffee cup have the same Euler number (fig. A.2). Mathematical [49] or more physical treatments [66] of the Euler number are in many textbooks. I will give only an outline here. The Euler number is: d X χ(M) ≡ (−1)i Bi .

(A.31)

i=0

The Bi are the Betti numbers. Bp is the number of independent closed p-surfaces that are not boundaries of a (p + 1)-surface. Also, by de Rham’s theorem, Bp is the number of independent closed p-forms modulo exact forms. For a closed manifold Bp = B4−p so χ = 0 for odd dimensional manifolds. For a simplicial complex, K, the Euler number is: X χ(K) ≡ (−1)r Ir r

where Ir is the number of r-faces in K. The equivalence of this with the Homology definition is the result of the Euler-Poincare Theorem. The Poincare-Hopf Theorem provides another interpretation of the Euler number: Consider any vector field on M which has only isolated zeros. The Euler number of a compact manifold is equal to the sum of the indices of the zeros. It is interesting that a topological quantity is equivalent to an analytic index. The following theorem also shows it to be equivalent to an integral.

Theorem A.12 (Gauss-Bonnet Theorem). For a compact manifold, χ is equal to the integral over M of the representative, Ω, on M of the Euler Class of its Tangent Bundle [18]. We will call this representative (give or take a numerical factor) the Euler density. The totally antisymmetric Levi-Civita symbol ǫa1 ...ad , with entries ±1, is a tensor w.r.t. the SO(2n) or Lorentz Group. For even dimension, d = 2n, we can construct the invariant: Ω≡

(−1)n a1 a2 Ω ∧ · · · ∧ Ωa2n−1 a2n ǫa1 ...a2n . (4π)n n!

(A.32)


93

It is the 2n-form (−1)n (4π)n n! Ω which we call the Euler density. The proof of the Gauss-Bonnet theorem in any even dimension is due to Chern [16]. It involves integrating over a section of the bundle of ortho-normal frames, known as the sphere bundle. On this manifold, Ω is a total derivative. There is a boundary at each singular point, contributing the index at each point- hence, by the Poincare-Hopf theorem, the proof follows. A.3.1 A useful property of the invariant polynomial Let M be a manifold with a Riemannian or Lorentzian metric g and a Levi-Civita connection. Let ω be the connection 1-form and Ω the curvature form. For dimM = 2n consider the integral Z f (Ω, ..Ω), f (Ω, .., Ω) = Ωa1 a2 ∧ ... ∧ Ωa2n−1 a2n ǫa1 ...a2n M

where ǫ... is the fully anti-symmetric symbol and ǫ1..2n = +1 and the integral is assumed to exist. The frame E is ortho-normal in the sense g(E a , E b ) = δ ab in the Riemannian case and g(E a , E b ) = η ab = diag(−1, 1..1) in the Lorentzian case. When g is Riemannian and M is compact and oriented f (Ω, ..Ω) represents the Euler class. The integral over M, normalised properly, gives the Euler number of M, as stated in the previous section. Let us now repeat the well known construction [18, 49] and show the following: Proposition A.13. under a continuous change of the connection, ω → ω ′ , f (Ω, ..Ω) changes by an exact form. Define ωt = tω + (1 − t)ω ′ . Call θ = ω − ω′ and note that θ=

d ωt dt

and for the curvature associated with ωt Ωt ≡ dωt + ωt ∧ ωt that

d Ωt = Dt θ, (A.33) dt where Dt is the covariant derivative associated with ωt . Then Z 1 Z 1 d ′ ′ dtf (dΩt /dt, Ωt , ...Ωt ) = f (Ω, .., Ω) − f (Ω , .., Ω ) = dt f (Ωt , .., Ωt ) = n dt 0 0 Z 1 Z 1 =n dtf (Dt θ, Ωt , ...Ωt ) = n dt df (θ, Ωt , ...Ωt ), (A.34) 0

0


94

where symmetry and multi-linearity of f have been used, as well as Dt Ωt = 0. If we define L(ω) = f (Ω, ..Ω) and ′

L(ω, ω ) ≡ −n we can write

Z

1 0

dtf (ω − ω ′ , Ωt , ...Ωt ),

L(ω) ≡ L(ω ′) − dL(ω, ω ′).

(A.35)

Now, assume that, for example, M is non-compact and without a boundary. If L(ω, ω ′) vanishes fast enough asymptotically, then Z L(ω) M

(assumed to exist) does not depend on ω. It is this property that makes L(ω) so interesting when, with a little modification, it is used as a Lagrangian for gravity for dimM > 2n, see section 2.5.


95

Fig. A.3: Why do bees make a hexagonal honeycomb? Is it because they have six feet or do they know something we don’t?

A.4 Honeycombs The study of the honeycomb has a long and illustrious history. It was studied by MacLaurin in the eighteenth century and Lord Kelvin in the nineteenth. In 1999 Thomas Hales [40] proved the two dimensional honeycomb conjecture: any partition of the plane into regions of equal area has perimeter which is at least that of the regular honeycomb tiling. Back in 36 B.C. Marcus Terentius Varro discussed the problem. He rejected the theory that bees built hexagonal structures because they have six feet. He suspected that the bees were great geometers. The mathematicians of the ancient world were highly sophisticated in geometry and the three regular tessellating shapes were well known. It was known that of the three the hexagon enclosed the most area. The three dimensional problem remains unsolved. It has been shown that the rhombic dodecahedron of the bees is not the solution. Lord Kelvin’s solution, while being better than the bees’ effort, was also proven false. Kelvin proposed a truncated octahedron, the Voronoi Cell of the body centred cubic packing of spheres. While this is minimum with respect to small deformations, it has recently been shown not to be the true minimum. In 1994 Weaire and Phelan found a counter example based on two different shapes. Kelvin’s proposal is still thought to be correct for identical shapes. The honeycomb illustrates key points of our study: ∗ The hypersurfaces intersect in such a way as to break up the bulk space into many regions. ∗ It is simplicially valent. The intersections have the minimum number (3) of hypersurfaces meeting there.

B. USEFUL FORMULAE B.1 The variational principle for a gravity theory For a general gravity theory, we have an action functional built by contracting products of the Riemann tensor with the metric tensor: S[g ab , ∂c g ab , ∂c ∂d g ab ]. The Euler variation of the action with respect to the metric tensor is: Z ∂L ∂L ∂L δS = δg ab ab + ∂c δg ab ab + ∂c ∂d δg ab ab ∂g ∂g ,c ∂g ,cd

(B.1)

where the comma denotes a partial derivative. We partially integrate the second term in (B.1): ∂L ab ∂L ab ∂L ∂c δg = ∂c δg − δg ab ∂c ab . ab ab ∂g ,c ∂g ,c ∂g ,c Now partially integrate the third term in (B.1): ! ∂L ∂L ∂L ∂c ∂d δg ab ab = ∂c ∂d δg ab ab − ∂d δg ab ∂c ab ∂g ,cd ∂g ,cd ∂g ,cd ! ! ∂L ∂L = ∂c ∂d δg ab ab − ∂d δg ab ∂c ab + δg ab ∂d ∂c ∂g ,cd ∂g ,cd

∂L ∂g ab,cd

!

.

We get H ab δg ab + ∂c V c , H ab

∂L ∂L = ab − ∂c ab + ∂d ∂c ∂g ∂g ,c

∂L V = δg − δg ab ∂d ab ∂g ,c c

(B.2)

ab

∂L ∂g ab,cd

!

∂L ∂g ab,cd

!

+ ∂d δg ab

,

∂L , ∂g ab,cd

or after further partial integration: ∂L V c = δg ab ab − 2δg ab ∂d ∂g ,c

∂L ∂g ab,cd

!

+ ∂d

∂L δg ab ab ∂g ,cd

!

.

B. Useful formulae

97

B.2 The variational principle for Lovelock gravity The action for General Relativity is Z Z S= Re + Lmat . M

M

The Lagrangian Lmat = Lmat e is due to the (unspecified) matter content such as gauge fields. The Euler variation w.r.t. g µν (neglecting boundary terms) Z ∂Lmat. δS = (Gµν − Tµν )δg µν e, Tµν e ≡ − ∂g ab M leads to the Einstein equations. Tµν is the Energy-momentum tensor, assuming Lmat is correctly normalised. For the Lovelock theory we have: Z Z S= LLovelock + Lmat . M

M

The Euler variation w.r.t. g µν (neglecting boundary terms) leads to: Z (Hµν − Tµν )δg µν e, δS = M

where Hµν is the Lovelock tensor. Similarly, if we have matter on some intersection I, we have a term in the action: Z ˜ mat (φ, γ)e, ˜ L (B.3) I

where φ represents the matter fields and γ the induced metric on I. e˜ is the induced volume element on I. δ L˜mat ≡ −T˜µν δg µν e˜

(B.4)

is the energy-momentum tensor on I. According to our formalism presented in chapter 4, T˜µν will be equal to the variation of the appropriate surface term. These more familiar expressions for the gravitational action principle are in terms of variation w.r.t. the metric. Throughout this thesis, I have used the vielbein language so it is useful to be able to translate between the two. We shall need these identities: δE b = δEµb Ecµ E c , δEµb Eaµ = −Eµb δEaµ , δg µν = 2η ab δEaµ Ebν .

(B.5) (B.6) (B.7)

B. Useful formulae

98

When one varies the gravitational part of the action with respect to the frame, the result is: δLG = δE b Eb = Eba δE b ea (We can always write it in this form by using equation A.22). Substituting (B.5): δLG = Eba Ecµ δEµb E c ea = Eba Eaµ δEµb e.

If we vary the action with respect to g µν , the variation of the matter part gives the Energy-momentum tensor: δLmatter = −Tµν δg µν e.

(B.8)

Using (B.7) and (B.6): δLmatter = −2Tµν η ab δEaµ Ebν e = 2Tba Eaµ δEµb e

= −2Tba δEaµ Eµb e.

(B.9)

Comparing (B.8) and (B.9), the Field equation is: Hba = Tba ,

1 Hba ≡ − Eba . 2

(B.10)

The variation of the surface terms in the gravity action are of the form: Z E˜ba δE b eã I

so similarly we get: ˜ a = T˜ a H b b

˜ a ≡ − 1 Eã . H b 2 b

(B.11)

˜ a the Junction Tensor of I. I shall call H b Of course, we can write this in a co-ordinate basis: Hνµ = Tνµ ,

1 Hνµ ≡ − Eaµ Eba Eνb . 2

(B.12)

It is important to remember this factor of −1/2 when equating the variation with the Energy momentum tensor. This has been used in equations (2.5), (6.7) and (6.25).

B. Useful formulae

99

B.3 Second fundamental form and tensors An embedded hypersurface, Σ, has a unique metric connection ω0 induced by the bulk metric. Throughout this section ι∗ is the pull back onto Σ acting on differential forms. The second fundamental form is: II = ι∗ (ω − ω0 ). The connection transforms non-tensorially but the difference of two tensors is tensorial: (ω − ω0 )ab V b = (D − D0 )V a for arbitrary vector V . Since the covariant derivative is tensorial, so the difference of two connections must also be. There is an important relation between the second fundamental form and the extrinsic curvature tensor: II ab = 2(n·n)n[a (∇b] nc )E c = 2(n·n)n[a K b] .

(B.13)

For a proof see e.g. Choquet-Bruhat et.al. [18]. Concentrate on a hypersurface {ij}. n{ij} = −n{ji} is the normal vector induced by i. I collect here some formulae [64, 63] b] [a b] [a ∗ ab ∗ ι (D{i} II{i|ij} ) = 2(n·n) ι D{i} n K{i} + n D{i} K{i} , {|ij}

a cb [II{i} , II{i} ]ab {|ij} = 2IIc ∧ II

a b c = −2 (n·n)K{i} ∧ K{i} + na nb K{i}c ∧ K{i}

{|ij}

,

(B.14)

a ι∗ (D{i} na{i|ij} ) = −K{i|ij} .

Above, i|ij denotes a quantity on {ij} induces by the bulk region {i}, in particular K{i|ij} 6= K{j|ij}. Also i is used as shorthand for i|ij when there is no ambiguity and n is shorthand for nij . Now, by ∼ I shall mean equality up to terms not explicitly involving a normal index. ab a b (B.15) ι∗ (Di II{i|ij} ) ∼ [II{i} , II{i} ]ab {|ij} ∼ −2 (n·n)K{i} ∧ K{i} {|ij} . Note that

2K a ∧ K b = Kca Kdb − Kba Kcb E c ∧ E d .

(B.16)

B. Useful formulae

100

At a hypersurface {ij} we can interpolate between the intrinsic curvature and the curvature of i: 1 Ω(t) = dωi + tdθ + [θ + tθ, ω0 + tθ] 2 1 = Ωi + tDij θ + t2 [θ, θ] 2 (t2 − 2t) = Ωi + tDi θ + [θ, θ]. 2

(B.17) (B.18)

Setting t = 1 and ι∗ θ = II: 1 Ωi = Ωij + (Di θ − [θ, θ]) 2 ι∗ Ωi ∼ Ωij − n·n Ki ∧ Ki .

(B.19) (B.20)

B.4 Decomposing the bulk Lagrangian In section 3.4, we re-wrote f (Ωn E d−2n ) − f (Ω0 E d−2n ) in such a way that the normal derivatives of the extrinsic curvature were in a total derivative term (equation 3.12). The other term was (3.13): Z 1 a Z =n dt (1 − t)θa1 a2 Ω(t)a3 ...a2n θb 2n+1 E b ea1 ...a2n+1 . 0

Now, for convenience, use adapted frames. Let E N be a unit normal vector and E i the tangent vectors. We make use of the following identities (see B.13 and B.20): θab = 2N [a K b]c E c + (normal), Ω(t)ij =

Ωij 0

(B.21)

− n·n K i ∧ K j + (normal).

(B.22)

There are factors of Ω(t) appearing in Z. It is easy to see that only the tangential components of the tangential form contribute. First note that if Ω(t) has a normal index, then θij ∧ θbj ∧ E b is proportional to E N ∧ E N = 0. Next note that if there are no normal indices then θi1 i2 ∧ · · · ∧ ei1 ···i2n+1 ∝ E N ∧ E N = 0. We get: Z =n

Z

1 0

i dt (1 − t) 2θN i2 Ω(t)i3 ...i2n θb2n+1 E b eN i2 ...i2n+1

+ θi1 i2 Ω(t)i3 ...i2n θbN E b ei1 ...i2n N . i

In the first term θN i1 is proportional to E N . In the second term, θb2n+1 ∧ E b will be proportional to E N . This means that the (normal) form part of Ωij does not contribute to Z. From (B.22), we see that there are no derivatives of the extrinsic curvature appearing in Z. This justifies the statement in section 3.4 that there are no second normal derivatives of the metric in Z.

B. Useful formulae

101

B.5 Proof of dη = 0 In this section, a short proof is given of Proposition 5.3. We start with the definition of Ω(t) and also the Jacobi identity [17]: 1 Ω(t) = d(x) ω(t) + [ω(t), ω(t)], 2 [[ω, ω], ω] = 0.

(B.23) (B.24)

From the above one can easily find the following identities. d(x) [ω(t), ω(t)] = 2[Ω, ω(t)], d(t) [ω(t), ω(t)] = 2[d(t) ω(t), ω(t)], 1 d(t) ω(t) + Ω(t) = d(F ) ω(t) + [ω(t), ω(t)]. 2

(B.25) (B.26) (B.27)

Also, from (B.25), we get the Bianchi identity for ω(t): D(t)Ω(t) = 0.

(B.28)

Let us show that ω(t) transforms as a connection ωα → g −1ωi g + g −1dg, X X ω(t) → C α g −1ωα g + C α g −1 dg α

= g −1

X α

−1

C α ωα

!

α

g + g −1 dg

= g ω(t)g + g −1dg.

(B.29)

Hence ω(t) is a connection and so the invariance property of f implies, for 2-forms ψ: X f (ψ1 ∧ ...[ω(t), ψi ]... ∧ ψn ) = 0 (B.30) i

Combining (B.25 − B.27):

d(F ) d(t) ω(t) + Ω(t) = [d(t) ω(t) + Ω(t), ω(t)]

(B.31)

and so our proposition 5.8 follows by the invariance property of the Polynomial (B.30). ∧n df d(t) ω(t) + Ω(t) =0

B. Useful formulae

102

Let us expand the polynomial. n ∧n X n Cl d(t) ω(t) + Ω(t) = l=0

=

n X l=0

p X α=1

d(t) ω(t)

!∧l

∧ Ω(t)∧(n−l)

(B.32)

(−1)l(l−1)/2 n Cl dtα1 ∧ · · · ∧ dtαl ∧ d(t) ω(t)α1 ∧ · · · ∧ d(t) ω(t)αl ∧ Ω(t)∧(n−l) .

The first term in the expansion evaluated at the 0-simplex {i} is just the Euler density in the interior of the region i. This, combined with the recursion relation (5.14) completes the proof by induction of Proposition 5.3. As a consistency check, we can see that the terms in this expansion reproduce the form of (5.10).

C. GLOSSARY Indices µ, ν, ..

Lower case Greek indices for d-dimensional space-time indices.

a, b..

Lower case Latin indices from the beginning of the alphabet for local Lorentz indices. Lower case Latin indices from the middle of the alphabet label the bulk regions. The hypersurface or intersection where the bulk regions i, j, . . . meet.

i, j, .. {ij}, {ijk}, ... Λ, Ξ

Upper case Greek indices label hypersurfaces.

Symbols Ck d

k times continuously differentiable or coefficient of ωk in expansion of ω(t). The dimension of a manifold.

d

Exterior derivative.

E

Co-frame field.

f

Invariant d-form, f (ψ) = ψ a1 ...ad ǫa1 ...ad .

fi

Partition of unity.

Ip

Non-simplicial intersection of co-dimension p.

s, si0 ...ip

Simplex (of dimension p).

t, tα

homotopy parameter(s).

η, σ

(bold Greek letters) Exterior differential forms.

ω

Connection 1-form.

Ω

Curvature 2-form.

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